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M. Rosenblum in [2] presented a most ingenious proof of the Fuglede and Putnam Theorems by means of entire vector valued functions [1, p. 59]. We will demonstrate that some curious properties of bounded Hilbert space operators can be derived from Rosenblum's argument and similar arguments. Throughout this text we mean by an "operator" a bounded linear transformation of a Hilbert space into itself. Given an operator A we mean by "exp A " the uniform limit of the series I+A +A 2/2 1 +A3/3! +A4/4! + * * * . We let A * denote the adjoint of the operator A, and let z* denote the complex conjugate of the complex number z. A "normal" operator is an operator which commutes with its adjoint. A critical fact in the Rosenblum proof is that given a normal operator A and any complex number z, exp (izA) exp (iz*A *) exp (izA +iz*A *) = exp (iz*A *) exp (izA), and this operator is unitary because i(zA +z*A *) is skew hermitian. Our first result states, among other things, that the converse is true; if the above equations hold for a fixed operator A and all complex numbers z, then A is normal.

Hilbert famously showed that polynomials in n variables are not too complicated, in various senses. For example, the Hilbert Syzygy Theorem shows that the process of resolving a module by free modules terminates in finitely many (in fact, at most n) steps, while the Hilbert Basis Theorem shows that the process of finding generators for an ideal also terminates in finitely many steps. These results laid the foundations for the modern algebraic study of polynomials. Hilbert's results are not uniform in n: unsurprisingly, polynomials in n variables will exhibit greater complexity as n increases. However, an array of recent work has shown that in a certain regime---namely, that where the number of polynomials and their degrees are fixed---the complexity of polynomials (in various senses) remains bounded even as the number of variables goes to infinity. We refer to this as Stillman uniformity, since Stillman's Conjecture provided the motivating example. The purpose of this paper is to give an exposition of Stillman uniformity, including: the circle of ideas initiated by Ananyan and Hochster in their proof of Stillman's Conjecture, the followup results that clarified and expanded on those ideas, and the implications for understanding polynomials in many variables., This expository paper was written in conjunction with Craig Huneke's talk on Stillman's Conjecture at the 2018 JMM Current Events Bulletin

This paper is about two arrangements of hyperplanes. The first --- the Shi arrangement --- was introduced by Jian-Yi Shi to describe the Kazhdan-Lusztig cells in the affine Weyl group of type $A$. The second --- the Ish arrangement --- was recently defined by the first author who used the two arrangements together to give a new interpretation of the $q,t$-Catalan numbers of Garsia and Haiman. In the present paper we will define a mysterious "combinatorial symmetry" between the two arrangements and show that this symmetry preserves a great deal of information. For example, the Shi and Ish arrangements share the same characteristic polynomial, the same numbers of regions, bounded regions, dominant regions, regions with $c$ "ceilings" and $d$ "degrees of freedom", etc. Moreover, all of these results hold in the greater generality of "deleted" Shi and Ish arrangements corresponding to an arbitrary subgraph of the complete graph. Our proofs are based on nice combinatorial labelings of Shi and Ish regions and a new set partition-valued statistic on these regions.

In this paper we give simple, sufficient conditions for the existence of the stochastic integral for vector-valued processes X with values in a Banach space E; namely, X is of class (LD), and the stochastic measure I X is bounded and strongly additive in L p E (in particular, if I X is bounded in L p E and c 0 ⊈ E) and has bounded semivariation. The result is then applied to martingales and processes with integrable variation or semivariation. For martingales the condition of being of class (LD) is superfluous. For a square-integrable martingale with values in a Hilbert space, all the conditions are superfluous. For processes with p-integrable semivariation or p-integrable variation, the conditions of I X to be bounded and have bounded semivariation are superfluous. For processes with 1-integrable variation, all conditions are superfluous. In a forthcoming paper, we shall extend these results to local summability. The extension needs additional nontrivial work.

In 1918 Pólya and Vinogradov gave an upper bound for the maximal size of character sums, which still remains the best known general estimate. One of the main results of this paper provides a substantial improvement of the Pólya-Vinogradov bound for characters of odd, bounded order. In 1977 Montgomery and Vaughan showed how the Pólya-Vinogradov inequality may be sharpened assuming the Generalized Riemann Hypothesis. We give a simple proof of their estimate and provide an improvement for characters of odd, bounded order. The paper also gives characterizations of the characters for which the maximal character sum is large, and it finds a hidden structure among these characters.

The first section of this paper provides an improvement upon known finiteness theorems for Riemannian submersions; that is, theorems which conclude that there are only finitely many isomorphism types of fiber bundles among Riemannian submersions whose total spaces and base spaces both satisfy certain geometric bounds. The second section of this paper provides a sharpening of some recent theorems which conclude that, for an open manifold of nonnegative curvature satisfying certain geometric bounds near its soul, there are only finitely many possibilities for the isomorphism class of a normal bundle of the soul. A common theme to both sections is a reliance on basic facts about Riemannian submersions whose A A and T T tensors are both bounded in norm.

In this paper it is shown that if MC = ( iS a 2 x 2 upper triangular operator matrix acting on the Hilbert space 7H E IC and if W(.) denotes the "Weyl spectrum", then the passage from w(A) U w(B) to w(MC) is accomplished by removing certain open subsets of w(A) n w(B) from the former, that is, there is equality w(A) U w(B) = w(MC) U G, where G is the union of certain of the holes in w(Mc) which happen to be subsets of w (A) n w (B). Let At and IC be Hilbert spaces, let 1C(7-, 1C) denote the set of bounded linear operators from 7t to IC, and abbreviate LC(QH,7) to C(H). When A E LC(t) and B E L(/C) are given we denote by Mc an operator acting on 1t E 1C of the form M A C) Mc:=( 0 ), where C E iC(!, 7-). The invertibility and spectra of Mc were considered by Du and Jin [5]. In this paper we give some conditions for operators A and B to exist an operator C such that Mc is Weyl, and describe the Weyl spectra of Mc. Recall ([7], [8]) that an operator A E 12(X, Y) for Banach spaces X and Y is called regular if there is an operator A' E L(Y, X) for which A = AA'A; then A' is called a generalized inverse for A. In this case, X and Y can be decomposed as follows (cf. [8, Theorem 3.8.2]): A-1(0) e A'A(X) = X and A(X) e (AA')-'(0) = Y. It is familiar ([6], [8]) that A E LC((, IC) is regular if and only if A has closed range. An operator A E LC(7-, IC) is called relatively Weyl if there is an invertible operator A' E 12(IC,7t) for which A = AA'A. It is known ([8, Theorem 3.8.6]) that A is relatively Weyl if and only if A is regular and A-1(0) A(H)', where means a topological isomorphism between spaces. An operator A E LQ(H, IC) is called left-Fredholm if it is regular with finite dimensional null space and rightFredholm if it is regular with its range of finite co-dimension. If A is both leftand right-Fredholm, we call it Fredholm. The index, ind A, of a leftor right-Fredholm Received by the editors November 21, 1997 and, in revised form, May 1, 1998 and March 10, 1999. 1991 Mathematics Subject Classification. Primary 47A53, 47A55.

We prove new results on the stability of the absolutely cont;inuous spectrum for perturbed Stark operators with decaying or satisfying certain smoothness assumption perturbation. We show that the absolutely continuous spectrum of the Stark operator is stable if the perturbing potential decays at the rate (1 + x) 3 -e or if it is continuously differentiable with derivative from the H6lder space C, (R), with any ae > 0. 0. INTRODUCTION In this paper, we study the stability of the absolutely continuous spect;rum of one-dimensional Stark operators under various classes of perturbations. Stark Schr6dinger operators describe behavior of the charged particle in the constant electric field. The absolutely continuous spectrum is a manifestation of the fact that the particle described by the operator propagates to infinity at a rather fast rate (see, e.g. [2], [12]). It is therefore interesting to describe the classes of perturbations which preserve the absolutely continuous spectrum of the Stark operators. In the first part of this work, we study perturbations of Stark operators by decaying potentials. This part is inspired by the recent work of Naboko and Pushnitski [14]. The general picture that we prove is very similar to the case of perturbatiorns of free Schr6dinger operators [9]. In accordance with physical intuition, however, the absolutely continuous spectrum is stable under stronger perturbations than in the free case. If in the free case the short range potentials preserving purely absolutely continuous spectrum of the free operator are given by condition (on the power scale) lq(x)l < C(1 + xKl)'E, in the Stark operator case the corresponding condition reads lq(x)l < C(1 + IXD)-2-E. If c is allowed to be zero in the above bounds, imbedded eigenvalues may occur in both cases (see, e.g. [14], [151). Moreover, in both cases if we allow potential to decay slower by an arbitrary function growing to infinity, very rich singular spectrum, such as a dense set of eigenvalues, -may occur (see [13] for the free case and [141 for the Stark case for precise formulation and proofs of these results). The first part of this work draws the parallel further, showing that the absolutely continuous spectrum of Stark operators is preserved under perturbations satisfying lq(x)l < C(1 + Jxl)-3-E, in particular even in the regimes where a dense set of eigenvalues occurs; hence in such cases these eigenvalues are genuinely imbedded. Similar results for the free case were proven in [9], Received by the editors April 14, 1997. 1991 Mathematics Subject Classification. Primary 34L40, 81Q10. ?)1999 American Mathematical Society 243 This content downloaded from 207.46.13.156 on Sat, 10 Sep 2016 04:57:39 UTC All use subject to http://about.jstor.org/terms 244 ALEXANDER KISELEV [10]. Our main strategy of the proof here is similar to that in [9] and [10]: we study the asymptotics of the generalized eigenfunctions and then apply Gilbert-Pearson theory [7] to derive spectral consequences. While the main new tool we introduce in our treatment of Stark operators is the same as in the free case, namely the a.e. convergence of the Fourier-type integral operators, there are some major differences. First of all, the spectral parameter enters the final equations that we study in a different way and this makes analysis more complicated. Secondly, we employ a different method to analyze the asymptotics. Instead of Harris-Lutz asymptotic method we study appropriate Prilfer transform variables, simplifying the overall consideration. In the second part of the work we discuss perturbations by potentials having some additional smoothness properties, but without decay. It turns out that for Stark operators the effects of decay or of additional smoothness of potential on the spectral properties are somewhat similar. It was known for a long time that if a potential perturbing Stark operator has two bounded derivatives the spectrum remains purely absolutely continuous (actually, certain growth of derivatives is also allowed, see Section 2 for details or Walter [21] for the original result). We note that the results similar to Walter's on the preservation of absolutely continuous spectrum were also obtained in [4] by applying different types of technique (Mourre method instead of studying asymptotics of solutions). On the other hand, if the perturbing potential is a sequence of derivatives of 6 functions in integer points on R with certain couplings, the spectrum may turn pure point [3], [5], [1]. In some sense, the 6' interaction is the most singular and least "differentiable" among all available natural perturbations of one-dimensional Schrbdinger operators [11]. Hence we have very different spectral properties on the very opposite sides of the smoothness scale. This work closes part of the gap. We improve the well-known results of Walter [21] concerning the minimal smoothness required for the preservation of the absolutely continuous spectrum and show that in fact existence and minimal smoothness of the first derivative is sufficient to imply absolute continuity of the spectrum. After submitting this paper, the author learned about the work of J. Sahbani [17], where, in particular, the results related to Theorem 2.1 of the present work are proven. Sahbani's results are slightly stronger than Theorem 2.1: the derivative of potential V'(x) is required to be bounded and Dini continuous in order for the absolutely continuous spectrum to be preserved. In addition, he shows that the imbedded singular spectrum in this case may only consist of isolated eigenvalues. The approach employed in [17] is an extension of conjugate operator method. 1. DECAYING PERTURBATIONS Consider a self-adjoint operator Hq defined by the differential expression Hqu =-u" -xu + q(x)u on the L2(-oo, oo). Let us introduce some notation. For the function f C L2 we denote by bf its Fourier transform

In this paper, the authors establish a multiplier theorem for Herz type Hardy spaces. Let Tm be a multiplier operator defined in terms of Fourier transforms by Tmf Q) = M(W for suitable functions f. It is well-known that there is a multiplier theorem for H1 (Rn) (see [FS]): if a > n/2 and (1) J ID3m(e)12d d 0, let us denote m6QW) = M(6077(0) It is easy to check that (1) is equivalent to (2) sup |m6||Ka 2(Rn) < 0, 6 2 where K2'2 2(R) is a non-homogeneous Herz space (see [BS]). By using some embedding relations on Herz spaces, A. Baernstein II and E. T. Sawyer [BS] weakened (2) into (3) sup iM6i||Kf1l (Rn) < 0, 6 where 0 < E < an2. In fact, this is just a special case of their theorem. In [BS], Baernstein and Sawyer showed that m is a bounded multiplier of H1 (Rn) under an even weaker condition than (3); see Theorem 3b in [BS, page 21]. By using the technique of Herz type Hardy spaces developed by the authors in [LY1]-[LY3] and [Y], in this paper, we shall first establish a multiplier theorem for the homogeneous Herz type Hardy space Hk n(1-1/q), 1(IRn) which is introduced by y ~~~q the authors of this paper in [LY1]. Then as simple consequences of this theorem, a multiplier theorem for the corresponding non-homogeneous version of the space Received by the editors April 13, 1995 and, in revised form, April 5, 1997. 1991 Mathematics Subject Classification. Primary 42B15; Secondary 42B30.

The aim of this paper is to present an example of a bounded balanced domain of holomorphy in C" (n > 3) with continuous Minkowski function that is not Kobayashi-finitely-compact. INTRODUCTION It is known [6] that if G c C" is a bounded Reinhardt-domain of holomorphy with 0 C G then G is finitely-compact with respect to (w.r.t.) the Carath6odorydistance cG, i.e., all cG-balls are relatively compact subsets of G w.r.t. the usual topology. In a more general case, if G = Gh = {z C Cn: h(z) < 1} is a bounded balanced domain of holomorphy with continuous Minkowski function h, then G is finitely-compact w.r.t. the Bergman-distance bG [4]. On the other hand, the continuity of h is a necessary condition for G = Gh to be finitely-compact w.r.t. the Kobayashi-distance kG [5, 1]. In this paper we give an example of a bounded balanced domain of holomorphy G = Gh C C3 with continuous h that is not kG-finitely compact and therefore, not cG-finitely compact. This answers a question formulated by J. Siciak in [7]. In particular, the example shows that, in general, there is no comparison of type bG < CkG for bounded balanced domains of holomorphy with continuous Minkowski function. DEFINITIONS AND STATEMENT We repeat some of the notions that will be needed in the sequel. Definition. A domain G c Cn is called balanced' iff whenever z C G and E C, 11?

In the present paper the reflexivity of a WOT-closed algebra generated by certain pairs of commuting shifts, which are not necessarily doubly commuting, is proved. In what follows B(H) denotes the algebra of all (linear bounded) operators on a separable complex Hilbert space H. Throughout the paper we simply write a subspace for a closed subspace of H and an-algebra (of operators) on H for a subalgebra of B(H) with the unit (the identity on H). If 5 is a subset of B(H) then W(() stands for the WOT (Weak Operator Topology)closed algebra generated by 5 while Lat5 denotes the lattice of all invariant subspaces for all operators from Y. For operators T1,..., Tn on H we write in the sequel I(T1, ..., Tn), Lat(T1, ..., T.), respectively, instead of ({ T,, . T.., T}) and Lat({T, ..., Tn}). AlgLat Y stands for the algebra of all operators on H which leave invariant all subspaces from Lat 59. An algebra v is called reflexive if and only if _v = AlgLat? . A family 5 c B(H) is called reflexive if the algebra 41(Y) is reflexive. An operator T E B(H) is called reflexive if 5 = { T} . Deddens [1] proved the reflexivity of an isometry using the Wold decomposition and a model for a shift. If a pair of commuting isometries is considered then a Wold-type decomposition need not exist [7]. Another difficulty is a lack of any model for a pair of commuting shifts in a general situation. In [3] the reflexivity of a pair { VI, V2J} of doubly commuting isometries (i.e. VJ/, V2 commute and so do Vi, J2*') was proved. Thus a pair of doubly commuting shifts is also reflexive. In this paper we prove, using a model from [3], the reflexivity of certain pairs of shifts, which are not necessarily doubly commuting. Let S denote a set of all pairs of nonnegative integers and G be a set of all pairs of integers. Following [3], we introduce some notations and definitions. Namely, a subset X c G is called a diagram if E E X, s E S imply q+s E X. The set of all diagrams is denoted by X. For 4 E G we define E.= {X c X: 4 E X}I. It is obvious that E. c E,+, (q E G, s E S). Let XF be a c-algebra Received by the editors July 31, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 47D25; Secondary 47B35.

Multiplication by the independent variable on H 2 ( R ) {H^2}(R) for R R a bounded open region in the complex plane C \mathbb {C} is a subnormal operator. This paper characterizes its minimal normal extension N N . Any normal operator is determined by a scalar-valued spectral measure and a multiplicity function. It is a consequence of some standard operator theory that a scalar-valued spectral measure for N N is harmonic measure for R R , ω \omega . This paper investigates the multiplicity function m m for N N . It is shown that m m is bounded above by two ω \omega -a.e., and necessary and sufficient conditions are given for m m to attain this upper bound on a set of positive harmonic measure. Examples are given which indicate the relationship between N N and the boundary of R R .

In this paper, we introduce the study of the Bohr phenomenon for a quasi-subordination family of functions, and establish the classical Bohr's inequality for the class of quasisubordinate functions. As a consequence, we improve and obtain the exact version of the classical Bohr's inequality for bounded analytic functions and also for $K$-quasiconformal harmonic mappings by replacing the constant term by the absolute value of the analytic part of the given function. We also obtain the Bohr radius for the subordination family of odd analytic functions., Comment: 12 pages; The article has been accepted to appear in Proceedings of the American Mathematical Soceity

Sidorenko’s conjecture states that for every bipartite graphH on {1, · · · , k} ∫ ∏ (i,j)∈E(H) h(xi, yj)dμ |V (H)| ≥ (∫ h(x, y) dμ )|E(H)| holds, where μ is the Lebesgue measure on [0, 1] and h is a bounded, nonnegative, symmetric, measurable function on [0, 1]2. An equivalent discrete form of the conjecture is that the number of homomorphisms from a bipartite graph H to a graph G is asymptotically at least the expected number of homomorphisms from H to the Erdős-Renyi random graph with the same expected edge density as G. In this paper, we present two approaches to the conjecture. First, we introduce the notion of tree-arrangeability, where a bipartite graph H with bipartition A∪B is tree-arrangeable if neighborhoods of vertices in A have a certain tree-like structure. We show that Sidorenko’s conjecture holds for all tree-arrangeable bipartite graphs. In particular, this implies that Sidorenko’s conjecture holds if there are two vertices a1, a2 in A such that each vertex a ∈ A satisfies N(a) ⊆ N(a1) or N(a) ⊆ N(a2), and also implies a recent result of Conlon, Fox, and Sudakov [3]. Second, if T is a tree and H is a bipartite graph satisfying Sidorenko’s conjecture, then it is shown that the Cartesian product T H of T and H also satisfies Sidorenko’s conjecture. This result implies that, for all d ≥ 2, the d-dimensional grid with arbitrary side lengths satisfies Sidorenko’s conjecture.

This paper introduces the notion of weak subordination: If F F and G G are meromorphic in the unit disc U \mathcal {U} , then F F is weakly subordinate to G G , written F > G F > G , provided there exist analytic functions ϕ \phi and ω : U → U \omega :\mathcal {U} \to \mathcal {U} , with ϕ \phi an inner function, so that F ∘ ϕ = G ∘ ω F \circ \phi = G \circ \omega . A class X \mathcal {X} of meromorphic functions is termed stable if F ≺ w G F \stackrel {w}{\prec } G and G ∈ X ⇒ F ∈ X G \in \mathcal {X} \Rightarrow F \in \mathcal {X} . The motivation is recent work of Burkholder which relates the growth of a function with its range and boundary values. Assume F F and G G are meromorphic and G G has nontangential limits, a.e. Assume further that F ( U ) ∩ G ( U ) ≠ ∅ F(\mathcal {U}) \cap G(\mathcal {U}) \ne \emptyset and G ( e i θ ) ∉ F ( U ) G({e^{i\theta }}) \notin F(\mathcal {U}) , a.e. This is denoted by F > G F > G . Burkholder proved for several classes X \mathcal {X} that ( ( ∗ ) ) F > G and G ∈ X ⇒ F ∈ X . \begin{equation}\tag {$(\ast )$}F > G \qquad {\text {and}}\quad G \in \mathcal {X} \Rightarrow F \in \mathcal {X}.\end{equation} The main result of this paper is the Theorem: F > G ⇒ F ≺ w G F > G \Rightarrow F{ \prec ^w}G . In particular, implication (*) holds for all stable classes X \mathcal {X} . The paper goes on to study various stable classes, which include BMOA, H p {H^p} , 0 > p ⩽ ∞ 0 > p \leqslant \infty , N ∗ {N_{\ast }} , the space of functions of bounded characteristic, and the M Φ {M^\Phi } spaces introduced by Burkholder. VMOA and the Bloch functions are examples of classes which are not stable.

For a connected open subset Ω \Omega of the plane and n n a positive integer, let B n ( Ω ) {\mathcal {B}_n}(\Omega ) be the space introduced by Cowen and Douglas in their paper Complex Geometry and Operator Theory. Our paper deals with characterizing the essential spectrum of an operator T T in B n ( Ω ) {\mathcal {B}_n}(\Omega ) for which σ ( T ) = Ω ¯ \sigma (T) = \bar \Omega and the point spectrum of T ∗ {T^ * } is empty. This class of operators forms an important part of B n ( Ω ) {\mathcal {B}_n}(\Omega ) denoted by B n ′ ( Ω ) {\mathcal {B}’_n}(\Omega ) . We use this characterization to give another proof of the result of Axler, Conway and McDonald on determining the essential spectrum of the Bergman operator. Let A n ( G ) = { S : T = S ∗ is in B ′ n ( G ∗ ) } {A_n}(G) = \left \{ {S:T = {S^ * }{\text {is}}\;{\text {in}}{{\mathcal {B}’}_n}({G^ * })} \right \} . We also characterize the weighted shifts in A 1 ( G ) {A_1}(G) .

In this paper we consider degenerate parabolic equations of the form ( ∗ ) ({\ast }) \[ β ( u ) t − div ⁡ A ( x , t , u , u x ) + B ( x , t , u , u x ) ∋ 0 \beta {(u)_t} - \operatorname {div} A(x,\,t,\,u,\,{u_x}) + B(x,\,t,\,u,\,{u_x}) \ni 0 \] where A A and B B are, respectively, vector and scalar valued Baire functions defined on U × R 1 × R n U \times {R^1} \times {R^n} , where U U is an open subset of R n + 1 ( x , t ) {R^{n + 1}}(x,\,t) . The functions A A and B B are subject to natural structural inequalities. Sufficiently general conditions are allowed on the relation β ⊂ R 1 × R 1 \beta \subset {R^1} \times {R^1} so that the porus medium equation and the model for the two-phase Stefan problem can be considered. The main result of the paper is that weak solutions of ( ∗ ) ({\ast }) are continuous throughout U U . In the event that U = Ω × ( 0 , T ) U = \Omega \times (0,\,T) where Ω \Omega is an open set of R n {R^n} , it is also shown that a weak solution is continuous at ( x 0 , t 0 ) ∈ ∂ Ω × ( 0 , T ) ({x_0},{t_0}) \in \partial \Omega \times (0,\,T) provided x 0 {x_0} is a regular point for the Laplacian on Ω \Omega .

In this paper we determine all the weakly compact derivations of a nest algebra. We also obtain necessary and sufficient conditions in order that a nest algebra admit compact derivations. Finally we prove that every compact derivation of a nest algebra d is the norm limit of finite-rank derivations. 1. Let _ be a Banach algebra and let X be a Banach s-module. By an X-valued derivation of _ we mean a linear mapping 8: sV-X with the property 8(ab) = a8(b) + 8(a)b for all a E :, b E .@. The derivation 8 is called compact if 8 is a compact operator between the Banach space s and X, and weakly compact if 8 is a weakly compact operator from s to X (i.e. 8(s) is relatively weakly compact in X, where SV is the unit ball of sV [4]). Let H be a complex Hilbert space, B(H) the algebra of all bounded operators on H and Y(H) = X the set of compact operators on H. In [7] Johnson and Parrott investigated derivations of a von Neumann subalgebra of B(H) with range contained in Y. They proved that in most cases such derivations are implemented by a compact operator. The general result was recently obtained by Popa [8] who proved that this is the case for all von Neumann subalgebras of B(H). Such derivations are known to be weakly compact [1]. On the other hand, in a series of papers [1, 9, 10], C. A. Akemann, S. K. Tsui, and S. Wright have determined the structure of all compact and weakly compact s-valued derivations of a C*-algebra s, and of all compact B(H) valued derivations of a C *-subalgebra of B(H). In this note we determine the structure of all s-valued compact and weakly compact derivations of a nest algebra S. In particular we prove that every compact derivation of a nest algebra s is the norm limit of finite rank derivations. We need the following result. LEMMA 1 [6]. Let H be an infinite dimensional Hilbert space. If 8 is a compact derivation of B(H) then 8 0. Received by the editors January 16, 1984 and, in revised form, July 1, 1985. 1980 Mathenmatics Subject Classification. Primary 47D25; Secondary 47B05.

In this paper several results on analytic diameters and analytic centers are obtained. We show that the extremal function for analytic diameter is unique and that there exist compact sets with many analytic centers. We answer negatively several problems posed by F. Miinsker. Introduction. In the theory of functions in the complex plane, many capacitytype set functions are defined in order to measure the size of sets. The notions of analytic diameter and analytic center were first introduced by Vitushkin [13], [14] in the way of developing his beautiful theory of rational approximation. Recently F. Minsker [9] obtained their several properties and posed in the last part of his paper [9, p. 93] open problems on them. In this paper we investigate certain properties of the extremal function for the problem of analytic diameters and construct instructive examples, most of which offer negative answers to Minsker's problems. In ? 1 we list the definitions and the notation which we use throughout this paper. In ?2 we state preliminary known results as a series of lemmas. In ?3 we derive a property of the extremal function, and as a corollary we show that the extremal function is unique. In ?4 we consider the case where the compact set consists of finitely many continua. In ?5 we give examples, most of which serve as counterexamples to Minsker's problems. 1. Definition and notation. Let K be a compact set in the complex plane, and let D = D(K) denote the unbounded component of the complement KC of K with respect to the Riemann sphere. By AB(D) we denote the Banach space of all bounded analytic functions in D, normed by the uniform norm 1 in D, that is 0110 = sup{ff(z)1: z E D} (1) for anyf E AB(D). Let eg(K) = {f E AB(D): IItIOO < l,f(oo) = 0). Any function which belongs to Gi(K) is often called an admissible function for K. The analytic capacity y(K) of K is defined by y(K) = sup{If'(oo)I:f E zie(K)} (2) It is well known that there exists a unique extremal function fo J e6(K) with fo(oo) = y(K), which is called the Ahlfors function of K [1]-[4], [7], [8]. Received by the editors January 11, 1980. A MS (MOS) subject classifications (1970). Primary 30A34, 30A44. ( 1981 American Mathematical Society 0002-9947/81/0000-041 4/$03. 50

In a recent paper F. Treves studied a model of complexes of pseudodifferential operators in an open set of R", establishing necessary and sufficient conditions for its semiglobal solvability. In the present paper, the authors give necessary and sufficient conditions for the global solvability of an analogous complex defined on an orientable, compact smooth manifold without boundary. 1. Introduction and statement of the theorem. Let B be a ^-dimensional (v > 1), compact, connected, orientable C°° manifold without boundary. We will denote by A a linear selfadjoint operator, densely defined in a complex Hilbert space H, which is unbounded, positive and has a bounded invsise^-1. (We may think of A as being, for instance, (1 - Ax)s on R", or a selfadjoint extension of \DX\.) We will use the scale of "Sobolev spaces" Hs (for s E R), defined by A : if s > 0, Hs is the space of elements u of H such that Asu E H, equipped with the norm \\u\\s = M^mIIo. where || ||q denotes the norm in H = H°; if s < 0, Hs is the completion of H for the norm \\u\\s = WA'uWq. The inner product in Hs will be denoted by ( , )s. Whenever s E R, m E R, Am is an isomorphism (for the Hilbert space structures) of Hs onto Hs~m. By H°° we denote the intersection of the spaces Hs, equipped with the projective limit topology, and by H~°° their union, with the inductive limit topology. Since for each s E R, Hs and H~s can be regarded as the dual of each other, so can i/00 and H~°° : with their topologies, they are the strong dual of each other. We denote by C°°(fi; H°°) the space of C°° functions in fi valued in H00. It is the intersection of the spaces C7(s;// ) (of the ^/'-continuously differen- tiable functions defined in fi and valued in H^ as the nonnegative integers y, k, tend to +00. We equip C°°(fi; Hx) with its natural C°° topology. We will denote by

Recently, in their remarkable paper Critical point theory for indefinite functionals, V. Benci and P. Rabinowitz gave a direct approach-avoiding finitedimensional approximationsto the existence theory for critical points of indefinite functionals. In this paper we develop under weaker hypotheses a simpler but more general theory for such problems. In the second part of the paper the abstract results are applied to a class of resonance problems of the Landesman and Lazer type, and moreover they are illustrated by an application to a wave equation problem. 0. Introduction. This paper is devoted to the critical point theory of strongly indefinite functionals in a real Hilbert space H. Here a strongly indefinite functional is a Cl-map H -* R which is neither bounded from above or below not even modulo a finite-dimensional subspace. Such functionals arise for example in the study of periodic solutions of the one-dimensional wave equation, in the study of periodic solutions of Hamiltonian systems of ordinary differential equations, in the existence theory for systems of elliptic equations, etc. To find the critical points of strongly indefinite functionals different methods were used. For example, Rabinowitz studied the restrictions of the functional to finite-dimensional subspaces, [1, 2], and then he tried to find the critical points of the original problem by passing to the limit for a suitable subsequence of the critical points of the restricted problems. Amann [3] reduced the indefinite functionals by means of a saddlepoint-reduction to a finite-dimensional problem. Of course, this is only possible under some strong hypotheses. Under certain assumptions the functional can be replaced by a dual variational problem, having some better properties. (See for example [4 or 5].) The first direct approach was given by Benci and Rabinowitz [6]. The essential in their paper is the observation that more detailed information about the special form of the deformations (a substitute for the gradient flow) associated to the functional is needed. They constructed deformations having special representations by solving appropriate differential equations approximately by time discretisation. This method has some disadvantages because one needs strong hypotheses in order to get the approximations uniformly on bounded sets. A main part of our paper will be the construction of flows and deformations having special representations by solving appropriate differential equations exactly Received by the editors March 30, 1981 and, in revised form, November 30, 1981. 1980 Mathematics Subject Classification. Primary 58E05; Secondary 47H25, 35L05.

In this paper, the concept of bounded slope variation, that of the convexity of a function with respect to an increasing function, and the Lebesgue-Stieltjes integral are used to further generalize a theorem of F. Riesz and to give a new proof based on a weaker hypothesis that a function which has bounded slope variation with respect to an increasing function m over [a, b] can be expressed as the difference of two functions each cf which is convex with respect to m on [a, b]. Introduction. In [4], F. Riesz proved that a necessary and sufficient condition that a function F defined on the interval [a, b] be the integral of a function of bounded variation on [a, b] is that F have bounded slope variation with respect to I over [a, b], where I is the function defined, for each x, by I(x) = x. In Theorem 3 of [1], Riesz's result was generalized using the Lane integral with respect to a continuous increasing function m on [a, b] instead of the Riemann integral. In Theorem 1 of this paper, it is shown that by using the Lebesgue-Stieltjes integral, the requirement that m be continuous on [a, b] may be deleted. A comparison of Definition I of this paper with Definitions 3 of [7] and 1 of [6] will reveal that what J. R. Webb and the author call "bounded slope variation with respect to m over [a, b]" is the same concept that A. M. Russell calls "bounded m-second variation on [a, b]" and, in case m(x) = x for each x, is the same concept that A. W. Roberts and D. E. Varberg call "bounded convexity on [a, b]". Hence Theorem 2 of this paper generalizes both Theorem 1.1 of [7] and Theorem 3 of [6]. DEFINITION 1. The statement that f has bounded slope variation with respect to m over [a, b] means that f is a function whose domain includes [a, b], m is a real-valued increasing function on [a, b], and there exists a nonnegative number B such that if {xi}).0 is a subdivision of [a, b] with n > 1, then n-I f(xi, 1) -f (xi) f(xi) -f (xi,) I < B. i-lm(xi+,)mX) M (xi)-( _) The least such number B is called the slope variation of f with respect to m Received by the editors October 7, 1976. AMS (MOS) subject classifications (1970). Primary 26A45, 26A5 1; Sccondary 26A42.

Introduction. Let D be an arbitrary domain on the Riemann sphere (that is, D is an open connected set on the sphere) and let p be a point of D. Let H '(D) denote the space of bounded analytic functions on D. We consider the problem of maximizing If'(p)l for fin H-(D) with If(z)lj< 1 for all z in D, and f(p) =0. In the case when D is the unit disc U and p =0, this is Schwarz's lemma and the solution is classical: the maximum is one and if If'(0) = 1, then f(z) = eiOz for some real constant 0. In the general case, the theory of normal families assures us that there is at least one fo in H (D) with foll? 1 I and fo(p) ? f'(p)I for all f in H (D) bounded by 1 which vanish at p. We call any such function extremal. Theorem 1 of this paper provides an elementary proof that there is only one extremal function. We thus add to our problem the question of investigating the properties of the extremal function. In the case when D is bounded by a finite number of disjoint analytic simple closed curves, the properties of the extremal function have been studied extensively (see [1] and [7]). In this case, the extremal function is known to be analytic over the boundary and to have modulus one there. We show that this is a local property and carries over to all points in the boundary of D where the boundary is sufficiently well behaved, no matter what the remainder of the boundary of D is like. In ?1 we also show that the extremal function is closely related to the problem of removable singularities for bounded analytic functions. In ?2 we discuss a related extremal problem and show that the properties of its solution imply that on many planar domains D, each function analytic and bounded by 1 on D may be approximated uniformly on compact subsets of D by a sequence of inner functions. We also briefly discuss extensions of the results to open Riemann surfaces. Finally, some remarks on previous work on Schwarz's lemma are in order. Carleson [3, pp. 78-82] and Havinson [8, Theorem 9] have previously shown that the extremal function is unique. Our proof differs substantially from each of theirs and is considerably more elementary. Furthermore, it does not make use of any of the results from the finitely-connected case. Also, our Theorems 2 and 3 and a weaker version of Theorem 4 have previously been proved by Havinson [8, Theorems 22, 28, and 20]; however, as in the case of Theorem 1, the proofs given here are appreciably simpler and more elementary. Indeed, part of the motivation for this paper was to find simpler and more transparent proofs of these results. In particular, it is only in the proof of Theorem 5, which describes the behavior

Introduction. In a recent paper [17] we obtained a spectral representation for the bounded normal operator solutions of a certain functional equation whose special cases include the semigroup equation (U,,, = U,LUV) as well as many others. In this note we continue our study of the abstract functional equation expressed in terms of algebras of measures. We shall consider therefore a mapping z -* UZ of a locally compact space Z into the set of unbounded normal operators on a Hilbert space. Since the operators are not necessarily bounded, the conditions imposed on the mapping z -* Uz vary somewhat from the conditions in the bounded case. The main result (Theorem 1) which we obtain here is a spectral representation analogous to the result valid in the bounded case [17; 18, Theorem 2, p. 8]. The methods which we use are suggested by the papers [12] and [13] of C. Ionescu Tulcea. For the general theory and terminology of unbounded operators we refer the reader to the works of E. Hille and R. Phillips [10], C. Ionescu Tulcea [11], F. Riesz and B. Sz.-Nagy [21]. Spectral representations for semigroups of unbounded operators are also discussed in A. Devinatz [3], A. Devinatz and A. E. Nussbaum [4; 5], R. Getoor [7], C. Ionescu Tulcea and A. Simon [14] and A. E. Nussbaum [19]. Theorem 1 of this paper contains the main part of Theorem 1 in [14] (and henceforth certain of the results given in [3; 4; 5; 7; 19]). The setting of the problem in algebras of measures is given in ??1, 2, and 5. Spectral families of Radon measures and direct sums thereof are summarized in ??3 and 4. The main result is stated and proved in ?6. In ?8 we apply the main theorem to derive a corollary which in the bounded case extends a result of S. Kurepa [16].

Suppose C is a compact subset of the plane having a piecewise smooth boundary AC. Let F(r, 0) be the Fourier transform, in polar coordinates, of the indicator function of the set C, where by the indicator function of C, we mean the function whose value on C is 1, and whose value on the complement of C is 0. In ?1 of this paper, we shall describe some relationships between geometric properties of C, and the asymptotic behavior of F(r, 0) as r -x- 00. In ?2, we shall give applications of the results of ?1 to some questions in the geometry of numbers. 1. If AC is sufficiently smooth, and has everywhere positive Gaussian curvature, it is known that the function 'D(6) = sup, r312IF(r, 0)1 is bounded on S' (cf. [1]). If AC has points of zero curvature, this need no longer be true (cf. [3]). The following, however, remains true: THEOREM 1. If AC is of class Cn + 3, for some integer n 1, and if the Gaussian curvature of AC is nonzero at all points of AC, with the possible exception of a finite set, at each point of which the tangent line has contact of order 1. Moreover, 'D(6) is always bounded, except in neighborhoods of those points of S' which, regarded as vectors, correspond to exterior or interior normals to AC at points of zero curvature. In a neighborhood of such a point 006 'D(6) is bounded by a multiple of [dist (6, 00)] -n - 1)/2n , where dist (6, 00) is the length of the smaller arc of S' connecting 0 and 6o, and nj is the largest order of contact which can occur between AC and its tangent line, at those points of AC at which the exterior normal is either 00 or- o REMARK. Theorem 1 has analogues in higher dimensions. I shall show in another paper, by different methods, that if C is a compact convex subset of Rn, whose boundary is analytic, and if F(r, 0) is the Fourier transform, in polar coordinates, of the indicator function of the set C, then supr r(n + 1)/21 F(r, 0)1 is of class LP on Sn1, for some p>2. If C is a polygon, the estimates are of a quite different character. THEOREM 2. Suppose C is a polygon. Then

The importance and usefulness of Taylor's theorem need not be dwelt upon here. We are interested in it for functions whose arguments and functional values belong to abstract spaces of the Frechet type. Consequently no Rolle's theorem can be even stated, and the proofs will be somewhat different from the usual proofs in the theory of numerically-valued functions. It is also to be expected that a slight strengthening of hypotheses will be required. However, it is not necessary to assume the existence of a uniformly continuous nth differential, as was done in the first announcement of these results.: It is sufficient that the function should have an nth variation (in the sense of Gateaux), with certain limitations on its discontinuities. The functions we discuss will be one-valued functions whose arguments x and functional values y belong to linear metric spaces X and ) respectively, briefly, functions F on X to S9. Linear metric spaces correspond to the spaces called by Frechet, "espaces (D) vectoriels." For the notations, postulates and fundamental propositions, we refer the reader to Part 1 of a paper by T. H. Hildebrandt and the author,? to avoid practically entire repetition of that section. No other parts of that paper are needed here. We shall consistently use the letters r, s, a, b, c to refer to real numbers, and the German t to refer to the real axis. The notation To will then refer to a set of open intervals of that axis. By the notation (ab) we shall mean the bounded closed interval of St with end points a and b. Whenever the space M) is required to be complete, this will be specifically mentioned. The form of remainder obtained in our Taylor's formula is a generalization of that given by Jordan|| and analogous at least to those obtained in

Introduction. Let G be a locally compact topological group which is unimodular, that is, the left and right Haar measures coincide. We propose, in this and subsequent papers, to investigate the structure of its "L2-system." By the "L2-system" we mean the (complex) Hilbert space formed from the complex-valued measurable functions f on G for which If| 2 is integrable with respect to the Haar measure, and with a product defined in terms of convolution. This product is not always defined (for that reason we call it an L2-system rather than an L2-algebra) but there are dense subsets (for example, L,nL2) on which multiplication always is defined so it is "almost" an algebra. Our theorems depend only on certain key properties of these L2-systems so we shall investigate general systems with these properties, calling them H-systems. We do this not only for generality but because when decomposing an L2-system into minimal parts we expect those parts to be H-systems though they need not be the L2-system of any group. This paper contains the definitions of an H-system and an L2-system; the proof that an L2-system is an H-system; theorems on idempotents and generation of ideals by idempotents; a functional calculus of self-adjoint elements in an H-system; a simple structure theorem for abelian H-systems; and an example of the L2-system of a special group, which has an interesting structure. Using the theorems of this paper we believe we shall be able to obtain the complete structure of H-systems--the ultimate theorems presumably asserting that every such system is a "direct integral" of simple systems, and that every simple system is a certain type of full matrix systembut with continuous rather than discrete matrices. At present these theorems are not proved but before even considering them carefully it seems necessary to develop the material of this paper as a foundation. One might ask why we choose to investigate this L2-system, in which multiplication is only partially defined, when there are other systems (for example, L1-the integrable functions) which are actually algebras and whose structure is still unknown. Our belief is that in spite of having only a partially defined multiplication our system may be easier to study, in somewhat the same way that certain questions are easier to answer about unbounded hypermaximal operators on Hilbert space than about bounded operators on a general Banach space. That there is a relation between that situation and ours is shown by the fact that left (or right) multiplication by a fixed element in an L2-system is, for many choices of the element, an unbounded hyper

The matrix A may be chosen to be a Jacobi matrix, that is, Ajj=0 for i-jI >1; and An,n+l> 0 may be required. If I is {0, 1, 2, * }, A is then determined uniquely, assuming the convention that any invariant subspace of A orthogonal to the 0th coordinate subspace will be ignored. (Assuming, that is, that if Ak,k?l =Ak1, k = 0 then in A = ((A ij)) indices will be let run up to k only.) For finite I, A is not in general determined uniquely. This paper gives a convenient canonical form for A. There is little trouble in including in this result the generalized sort of moment problem introduced by Nagy [6], where un above are in 63, the set of bounded self-adjoint operators on a Hilbert space SC. One virtue of the generalized problem is its application to the classical problem; see Proposition 4 below. Accordingly some of the facts outlined in the preceding paragraph may as well be proved for the generalized problem; this is done in ?2, which in fact is essentially a recitation in the wider setting of the proof for the classical case. The main result is in ?3. The following sections mostly examine the classical problem in light of it. I have found the canonical form handy for getting numerical bounds on momenits, but will not discuss this use further. Throughout the paper I use the notation a--1 -a, where a may be a number or an operator. If it is an operator this requires some understanding as to the space on which it operates, but I think I have avoided amlbiguity.

The first chapter of this paper concerns itself with questions of uniform boundedness of sets of points in a Banach space and sets of functionals on a Banach space, as well as with a group of closely related resonance theorems. A well known example coming under this heading is the theorem of Toeplitz [36] t stating that supm Z I= a,.I < oo providing 77m =Zn-, -amntn converges whenever S, does. Another is the theorem of Hahn [12 ] stating that if an arbitrary continuous function has the partial sums of its Fourier expansion, with respect to an orthonormal sequence of bounded functions (n, essentially bounded, then the sequence f ZX= 1c^,=(x)co,(t) I dt is also essentially bounded. Still another is the theorem stating that if the adjoint of an everywhere defined transformation between Banach spaces is everywhere defined, then the transformation is continuous. This was proved, at least for Hilbert space, by von Neumann [21 ], Stone [34 ], Tamarkin ( [34], p. iv), and Stone and Tamarkin [35 ] and is probably not usually thought of as a theorem on uniform boundedness. Most of the results w-e have in the first chapter are new, others have been proved only in special cases, and some are well known but the proofs heretofore given have been different. Previous methods for discussing questions of uniform boundedness divide themselves into three groups (i) those associated with the names of Lebesgue [18], Banach [2], Hahn [11], and Hildebrandt [13 ]; (ii) those characterized by the elegant and direct use of the Baire category theorem as in the works of Banach [1], Saks [31], Saks and Tamarkin [32], and others; and (iii) those employed in a recent theorem of Gelfand [9] the proof of which is closely related to that of the category theorem. The second chapter of this paper is concerned with cases where a weak limiting process implies a strong one. These cases seem to be rather rare but probably more will come to light in the future. The theorem quoted above, stating that the existence of the adjoint implies the continuity of the function, belongs to the class of questions discussed in Chapter II as well as those discussed in Chapter I, while its analogue (Theorem 42) in the Boolean ring of

Introduction. By "the space of continuous functioins" we mean the Banach space C of real continuous functions on a compact Hausdorff space X. C, its first dual, and its second dual are not only Banach spaces, but also vector lattices, and this aspect of them plays a central role in our treatment. In ??1-3, we collect the properties of vector lattices which we need in the paper. In ?4 we add some properties of the first dual of C-the space of Radon measures on X-denoted by L in the present paper. In ?5 we imbed C in its second dual, denoted by M, and then identify the space of all bounded real functions on X with a quotient space of M, in fact with a topological direct summand. Thus each bounded function on X represents an entire class of elements of M. The value of an element f of M on an element ,i of L is denoted as usual by f(,u). If f lies in C then of course f(,u) =,u(f) (usually written f fd,u), and thus the multiplication between M and L is an extension of the multiplication between L and C. Now in integration theory, for each ,uEL, there is a standard "extension" of ffd,u to certain of the bounded functions on X (we confine ourselves to bounded functions in this paper), which are consequently called ,i-integrable. The question arises: how is this "extension" related to the multiplication between M and L. ??6-8 (and ?12) are devoted to this question. We show that given a bounded function f on X the equivalence class of M represented by f contains two distinguished elements, which we denote by f* and f*, such that for every Radon measure ,u, f*(,u) = f*fd,u and f*(,u) = f*fdM, where f* and f* are the lower and upper integrals of f with respect to ,u. Moreover, if f is integrable for every Radon measure, then f* and f* coincide, and we have a unique element whose value on each ,u is ffd4. In ?9 we study order-convergence in M. It is in this area that Af offers its most striking immediate gain in clarity over consideration of only the bounded functions oil X. We cite two examples. The first example: The characteristic functions of finite subsets of the real initerval 0< x < 1 form a directed system which order-converges to the constant function 1, while their Lebesgue integrals, being all zero, do not converge to the Lebesgue integral of 1. In M, however, this directed system order-converges to an element different from 1, and the above unsatisfactory situation disappears. In point of fact, order-convergence is perfectly well-behaved in M: if a directed system order converges, its values on every Radon measure converge

By using the results of [6], it is proved that for an extensive class of increasing functions h, ( ∗ ) ∑ 1 ≦ d ≦ x μ ( d ) d h ( x d ) ∼ x h ′ ( x ) as x → ∞ \begin{equation}\tag {$*$}\sum \limits _{1 \leqq d \leqq x} {\frac {{\mu (d)}}{d}h\left ( {\frac {x}{d}} \right )} \sim xh’(x)\quad {\text {as}}\;x \to \infty \end{equation} where μ \mu denotes the Möbius function. This result incidentally settles affirmatively Remark (iii) of [6], and refines the Tauberian Theorem 2 of that paper. It is also shown that one type of condition imposed in [6] is necessary to the truth of the cited Theorem 2, at least if some sort of quasi-Riemann hypothesis is true. Nevertheless, examples are given to show that on the one hand ( ∗ ) ( ^\ast ) may be true for functions not covered by the first theorem of this paper, and on the other that some sort of nonnaïve condition on a function h is necessary to ensure the truth of ( ∗ ) (^\ast ) .

The Fourier transform of an unbounded spectral distribution is studied: An explicit integral representation is obtained; connections are drawn to the associated generalized scalar operator. It is proved that every generalized pseudo-hermitian operator is the infinitesimal generator of a temperate CO group. Introduction. In [6] the author introduced a theory of unbounded spectral distributions in Banach spaces, and a corresponding theory of the generalized scalar operators which they represent. Properties of these objects were studied, culminating in a spectral mapping theorem [6, Theorem 4]. In this paper we study the Fourier transform of an unbounded spectral distribution, deriving an explicit integral representation, as well as growth estimates at infinity. The case of real support is considered in some detail, leading to the proof that every generalized pseudo-hermitian operator (i.e., generalized scalar with real spectrum) is the infinitesimal generator of a temperate group. This result generalizes the corresponding result for bounded operators proved in [5]. In this paper, all definitions are as in [6]. Integral representation of the Jourier transform. THEOREM I (EXTENSION OF SPECTRAL DISTRIBUTIONS). Let T be a spectral distribution [6, Definition 2] in a Banach space X. Then (a) for each fuinction ffromn the space (R2) of C' functions wt ith bounded lerivatives, and for each x E X, limit,, Tf pnx exists, where {c p} is a sequence of test flnctions as in [6, Definition 2(c)]. (b) If we define Tfx=Jimitn Tfp,,x for all x E X, then Tf is in the space Y(X) of all bounded linear operators on X, and the correspondence f--*Tf is a continuous linear mapping of X(R2) into f(X). (c) 7 T= Tf T, for al/f, g E .6(R2). (d) The operator Tf is independent of the sequence {p7J PROOF. (a) Define a double sequence of functions m-f(%-pm). Then {y form i X(R 2)-bounded subset of ?/ (R2) (cf. [3, p. 91 ] for the Received by the editors October 20, 1971. AAIS 1970 sub/jecr ch ssifications. Primary 47A65, 47A60, 47B40; Secondary 46F99. ? American Mathematical Society 1972

It is shown here that the mixed initial-boundary value problem for the Euler equations for ideal flow in bounded domains of R3 has a unique solution for a small time interval. The existence of a solution is shown by converting the equations to an equivalent system involving the vorticity and applying Schauder's fixed point theorem to an appropriate mapping. Introduction. The purpose of this paper is to prove the existence and uniqueness of classical solutions to the mixed initial-boundary value problem for the Euler equations for an ideal incompressible fluid in bounded domains of R3 for a small time interval. This has been shown by Ebin and Marsden [3] for compact Riemannian manifolds possibly with boundary. In this paper we obtain a new proof by converting the equations to an equivalent system involving the vorticity and solving this by using the Schauder fixed point theorem. The technique is similar to that of Kato [4] where he proved existence of ideal flow in bounded domains of the plane for an arbitrary time interval. The author also used a similar method to obtain an existence result for ideal flow in all R3 by showing that a solution is the limit of Navier-Stokes flow as the viscosity goes to zero [7]. Existence of a solution for a short time in all R3 was originally shown by Lichtenstein [5]. The Euler equations for ideal incompressible flow in a domain D in R3 with boundary bD are (Et) ar/at + (r grad)r =-grad P + B, V * U = O, with constraints U n = 0 on bD and r(0) = A, where x is a point in D U bD; t E [O, T], Iv(X, t) is the velocity vector, P(x, t) is the (scalar) pressure, B(x, t) is the external force field vector, n(x) is the outward normal vector to bD and A(x) is the initial velocity vector. By formally computing the curl of (E') we get the system Presented to the Society, January 18, 1972 under the title The existence and uniqeness of nonstationary idealflow in bounded domains in R3; received by the editors October 12, 1971 and, in revised form, March 20, 1972. A J1S (MOS) subject classifications (1970). Primary 35F25, 35F30, 35Q05, 76C05.

Introduction. This paper is mainly concerned with the characterization problem for bounded spectral operators (see [5]). We restrict the discussion to scalar type operators with real spectrum, which we call pseudo-hermitian operators (p. h.). The criteria are given in terms of certain properties of the exponential group generated by the operator. The simplest criterion of this kind is for Hilbert space: a bounded linear operator S is p.h. if and only if || eitS| ? M < oo for all real t. This statement is false for Banach spaces, and even for reflexive Banach spaces. In the latter case, we give additional necessary conditions on the group eitS, which together with its uniform boundedness are both necessary and sufficient for S to be p.h. Another useful criterion is the following: a bounded linear operator S on a reflexive Banach space is p.h. if and only if for everyf in L1(R) (R the real line), we have II fRf(t)e itsdt || _ M || f l,,, where the norm on the left is the operator norm, f is the Fourier transform off and || * is the sup norm. Unlike the general theorems in [5] about the spectrality of an operator, the characterizations obtained in this paper are easily applied to analytically given operators on concrete Banach spaces. An example is discussed in ?4. Applications are discussed at the end of ?5. ?2 deals with an improvement of a theorem of Foguel [7] about the resolutions of the identity of sums and products of commuting spectral operators. ?1 contains preliminaries. The characterization problem for spectral operators is discussed in ??3 and 5.

Introduction. This paper is concerned with extremal problems in the family of bounded analytic functions in a multiply-connected domain D, and it is concerned with extremal problems for the mean modulus f cIf Ids of meromorphic functions f in D taken over the boundary C of D. These two types of problems are shown to be closely related, and solutions are obtained simultaneously for both types by a method of contour integration. An altogether analogous method was exploited by Grunsky in his thesis [8](2) for the investigation of schlicht functions in a multiply-connected domain. Thus it is interesting to remark that the fundamental distortion theorems of schlicht conformal mapping theory have been developed by Grunsky by a method which we are able to apply here to obtain the fundamental distortion theorems for bounded functions. It will appear, then, that the generalization of Schwarz's lemma to multiply-connected domains and the generalization of the Koebe distortion theorem can be carried out by a unified technique(3). We shall find in addition to this that, while the recent papers of Bergman and Schiffer [5, 16] have developed a close relationship between the theory of schlicht canonical mappings and the Bergman kernel function [3], we are able to develop here a relationship between the theory of bounded functions and the Szego kernel function [19]. Thus the Szego kernel function does for the theory of distortion of bounded functions what the Bergman kernel function does for the theory of distortion of schlicht functions. We point out that both these kernel functions are actually differentials, and that in the Szego case one is dealing with length and in the Bergman case one is dealing with area. Thus the mean modulus fc f I ds, or ffD IfI 2dxdy, should be thought of as a length, or area, and not as a mean modulus. All these remarkable relationships are brought to light by using the simple boundary relations satisfied by the classical domain functions, such as

The two main results in this paper are analogues of the Stone-Weierstrass theorem for real-valued functions, obtained by using different function space topologies. The first (Theorem 2.3) is a Stone-Weierstrass theorem for unbounded functions. The second (Theorem 3.6) is a theorem for bounded functions; it is stronger than the usual theorem because the topology is larger than the uniform topology. In fact it is the strongest possible theorem in a sense made precise in Theorem 3.5. However, the usefulness of the result is, in both cases, limited by the fact that the topologies used are far from being as well behaved as the uniform topology. Some specific defects of these topologies are discussed in Proposition 2.4, Proposition 3.9, and Example 3.10. (The topologies are defined in ?1.) The formulation of the Stone-Weierstrass theorem on which the analogues in the present paper are based is the following: If X is a completely regular topological space, then X is compact iff every strongly separating subalgebra of C*(X) is u-dense, where u denotes the uniform topology. In [5] Lorch has studied a more general class of function algebras than those of the form C*(X). All of the results for bounded functions in this paper are formulated in this more general setting (see ?1).

THnORUM I. If the points z1, Z2, Z3 vary independently and have circular regions as their respective loci, then the locus of the point Z4 defined by the real constant cross ratio X=(Zl, Z2, Z3, Z4) is also a circular region. It is the purpose of the present paper to consider generalizations of and other results related to Theorem I, primarily the determination of the locus of the point Z4 defined as in Theorem I, when zi, Z2, Z3 vary independently so as to have certain prescribed loci. Thus one may raise the following question: If two variable points lie respectively on two fixed circles, where does the mid-point of their segment lie? The answer is contained in Theorem VI. Or again, if the loci of the points zi, Z2, Z3 are regions each bounded by a number of circles, what can be said of the locus of z4? The answer is given by Theorem VII. All the results proved concerning such loci as these can be interpreted in terms of the roots of the jacobian of two binary forms. Such an interpretation is given in ?12. The chief method used in the present paper to determine the locus of Z4 in any given case is that suggested in I (pp. 102, 103, footnote), namely the determination of the locus of z4 when z1 and Z2 are held fast and Z3 varies over its locus; the determination of the locus of this locus of z4 when z1 is kept fixed but Z2 varies over its locus; and finally the determination of the locus of this new locus of Z4 when z1 varies over its prescribed locus. In the present paper this method is carried through geometrically. The same method has been used by Professor A. B. Coblet to determine analytically the locus of Z4 in Theorem I

In this paper we shall confine ourselves to a two-dimensional euclidean space which we shall denote by the symbol S. R. L. Mooret has given the following extension of a theorem by Zoretti:t If K is a bounded maximal connected subset of a closed point set M and does not separate space, and e is a positive number, then there exists a simple closed curve containing no point of M, such that the interior of this curve contains K but contains no point whose distance from K is greater than e. Zoretti's conclusion is weaker than Moore's in that it says nothing about the points on the interior of the curve surrounding K. On the other hand, Moore finds it necessary to use a stronger hypothesis than Zoretti's; namely that K does not separate the plane. In this paper we consider among other things the question of the conditions involved in a combination of Zoretti's hypothesis and Moore's conclusion, and show in Theorem I that a result analogous to

for almost all t. In this definition it is assumed, of course, that the integral on the right-hand side of (1.2) is in the domain of A. In a recent paper [6], Friedman and Shinbrot have studied the equation (1.1) even in the more general case where xo and A depend on t and -, respectively. They proved theorems of existence, uniqueness, differentiability and asymptotic behavior of solutions. They also constructed fundamental solutions and derived asymptotic bounds for them. We recall [6] that if xo is in the domain of A4, for some ,u>0, then the solution of (1.1) is given by S(t)xo. The purpose of the present paper is to derive monotonicity theorems for solutions of (1.1). We shall generalize some of the monotonicity theorems of Friedman [2] (see also [4]) from the case X= R1 (R1 the one-dimensional Euclidean space) to the case where X is any Banach space. In ?2 we give some auxiliary results. These results are concerned with Volterra equations in one-dimension (i.e., X=R1). In particular, we study the behavior of the solutions with respect to a certain parameter. In ?3 we give an integral formula for S(t) in case A is a bounded operator. For A unbounded, we construct a fundamental solution as a limit of fundamental solutions corresponding to the bounded operators A(I+ A/n) -1. We prove that the fundamental solution coincides with the fundamental solution of [6, Chapter 1]

Introduction. In Cameron's paper [1] there appears a "rectangle formula" (there denoted by Theorem 1) by means of which the Wiener integral of F[x(.)], defined on the space of continuous functions on [0, 1 ] such that x(O) = 0, can be approximated by means of an n-fold Riemann integral provided F is sufficiently smooth and dominated by a suitable integrable functional. The formula employs a particular C.O.N. set of functions {ai(s) } 1 (the odd harmonic cosine functions.) A generalization of this formula appears in Cameron's unpublished notes and appears with proof in the author's [2], (there denoted by Theorem 2). The generalization allows for the use of an arbitrary C.O.N. set {ai(s) } of B.V. but suffers the defect that it is not known to hold except for bounded functionals. It is the purpose of this paper to show that if the {ai(s) }'s of this generalization are restricted to certain Sturm-Liouville sets of functions (among which sets is included the odd harmonic cosine functions), then the restricted generalization is applicable to suitably dominated (not necessarily bounded) functionals. Let w(s) >0 have continuous fourth derivative on [0, 1] and let {As} and {ai(s) } be the sets of characteristic numbers and normalized functions of the Sturm-Liouville problem: { [l/w(s) ]x'(s) }' +-Xa(s)=0, a'(O)=ca(1)=0 on [0, 1]. Let C be the space of continuous functions x(s) on [0, 1] such that x(O) = 0, let fli(s) =f'ai(t)dt and for x ()EC let ci = foea (s)dx (s): i = 1, 2, 3, * * (c, is of course a functional of x(.)). The main result of the paper is the following.

In this paper is presented a theory of unbounded operators which admit spectral distributions. The compactness of the support in the bounded case has been replaced by certain growth conditions at infinity. Properties of these operators are studied, including the existence of a Spectral Mapping Theorem. Inttoduction. In [2], Foias introduced the class of bounded operators in Banach spaces which in lieu of spectral measures admit spectral distributions. He proved certain basic properties of these "generalized scalars", such as the Spectral Mapping Thorem. Further investigations of this class of operators and of spectral distributions were made by various authors in [1], [2], [4], [5], [6],1 [8], and [10]. In [5], the present author studied the class of generalized scalars with real spectrum, which were called generalized pseudo-hermitian (g.p.h) operators. These operators were characterized by the elementary condition that the real exponential group generated by such an operator have polynomial growth in the uniform operator norm.2 In attempting to study the spectral properties of the infinitesimal generators of CO-groups with polynomial growth, the author has been led to consider unbounded generalized scalar operators. In this paper is presented a theory of unbounded spectral distributions and of the unbounded generalized scalar operators which they represent. The compactness of the support in the bounded case has been replaced by certain growth conditions at infinity sufficient to insure that the definitions are correct and the resulting class of operators well behaved, in that they generalize both the bounded generalized scalars and the unbounded normal operators in Hilbert space. Some properties of the class are discussed, culminating in a Spectral Mapping Theorem. 1. Basic definitions. DEFINITION 1. Let T be a mapping from the space -9(R2) of complex test functions into the space Y(X) of linear continuous operators on a Received by the editors April 6, 1971. 4MS 1970 subject classifications. Primary 47A65, 47A60, 47B40; Secondary 46F99. I Also considers unbounded operators. 2 Cf. Theorem 2 of [101. (? American Mathematical Society 1972

The perturbation theory of linear operators concerns itself with the eigenvalues and eigenelements of a variable operator. If T(e) is an operator depending on a real parameter e, the relation between the eigenvalues and eigenelements of the perturbed operator T(e) (e%0) and those of the unperturbed operator T(O) is of particular interest. This theory finds considerable application in the field of quantum mechanics. In such applications the eigenvalues of an operator represent certain quantities which, by their very nature, must be real. It is therefore not surprising to find that emphasis has been given to self-adjoint operators. The first person to handle systematically the regular perturbation of self-adjoint operators was Franz Rellich [7].2 By a different attack on the problem, Bela v. Sz. Nagy was able to improve on Rellich's theorem [5]. It was the conjecture of Frantisek Wolf that certain of these results would hold for normal operators-a class of operators which includes the self-adjoint. This paper substantiates that conjecture. Since the writing of this paper, Professor Wolf has rendered its main result a consequence of a general theory of analyticity for operators in a Banach space [9, section 6.4]. The proof given here exploits the relationship between a normal operator and a pair of self-adjoint operators. A bounded linear transformation of an abstract Hilbert space i into itself will be called an operator on t. The number X(e), the element f(e) of &, and the operator T(e) on I will be called regular on the interval I e

In this paper we study the spectra of conservative matrices and show that the spectrum of any Hausdorff method is either uncountable or finite. In the latter case it is shown that the spectrum consists of either one point or two points. We obtain the sharpest possible Mercerian theorem for Euler methods. We also get some results about the location of conservative matrices with respect to the maximal group of invertible operators. Let wo represent the space of all sequences and let c (co or m) represent the space of convergent sequences (null sequences or bounded sequences). Let B(c) represent the Banach algebra of bounded operators on c with the usual norm topology. Let A=(ank) be any infinite matrix and let, for an x in ow, Ax denote the transformed sequence, provided it exists. Let CA = {x E a): Ax E c}. A is called conservative (coercive) if CA 2 C (CA 2i m). For the properties of conservative (coercive) matrices and Hausdorff methods the reader might refer to [1]. In this paper we first obtain information about the spectrum of a conservative Hausdorff method and use this information to prove that a Hausdorff method is coercive if and only if it is a scalar multiple of Ho which we will define later on. This generalizes the result a-bout right finite Hausdorff methods which appears in [3]. Incidentally we show that the answers to Questions 1 and 4(i) of Rhoades [4] are in the affirmative. Then we discuss the spectral aspects of Mercer's theorem for Euler's methods. These results show that the discussion about the location of Euler's methods in [4] is not correct. Finally we list two open problems about the structure of the spectra of conservative matrices. We wish to thank Professor B. E. Rhoades for many fruitful conversations. A matrix A is called triangular if ank=0 whenever k>n and a triangullar matrix A is called a triangle if ann$O for any n. A denotes the Banach algebra of triangular matrices in B[c], G denotes the group of invertible operators in A and aG the boundary of G in A. The following theorem about coercive triangular matrices has been essentially proved in [5]. Presented to the Society, January 17, 1972; received by the editors November 18, 1971 and, in revised form, February 24, 1972. AMS 1970 subject classifications. Primary 40H05, 40G05; Secondary 47B99. ( American Mathematical Society 1972

If ? is a function whose domain is a subset E of the set of curvilinear convergence off, then ? is called a boundary function for f if, and only if, for each x E E there exists an arc y at x such thatf(z) -+ ;(x) as z -* x along y. Let S be another metric space. We shall say that a function ? is of Baire class < 1(S, M) if (i) domain ? = S, (ii) range ? c M, and (iii) there exists a sequence {0n} of continuous functions, each mapping S into M, such that n, -* 0 pointwise on S. We shall say that ? is of honorary Baire class < 2(S, M) if (i) domain ? = S, (ii) range ? c M, and (iii) there exists a countable set Nc S and there exists a function b of Baire class _ 1(S, M) such that ?(x) = +(x) for every x E S-N. It is known that iff is a continuous function mapping D into the Riemann sphere, then the set of curvilinear convergence off is of type Fa,, and any boundary function forf is of honorary Baire class _ 2(C, Riemann sphere). (See [3], [4], [5], [6], [9].) J. E. McMillan [6] posed the following problem. If A is a given set in C of type F,6, and if ? is a function of honorary Baire class _2(A, Riemann sphere), does there always exist a continuous function f mapping D into the Riemann sphere such that A is the set of curvilinear convergence of f and ? is a boundary function for f? The purpose of this paper is to give an affirmative answer to McMillan's question. However, the corresponding question for real-valued functions remains open. (See Problems 1 and 2 at the end of this paper.) In proving our result, we first give a proof under the assumption that ? is a bounded complexvalued function, and we then use a certain device to transfer the theorem to the Riemann sphere. As we shall indicate in an appendix, the same device can be

Introduction. In the study of uniform approximation to a function of a complex variable by polynomials or by bounded analytic functions, Lipschitz conditions have proved extremely useful in relating degree of approximation to continuity properties of the functions approximated. Parts of this theory are analogous to the older study (S. Bernstein, D. Jackson, de la Vallee Poussin, Montel) of approximation to real periodic functions by trigonometric sums, where Lipschitz conditions have also proved useful. Hardy and Littlewood [2, p. 633 ](2) first pointed out that trigonometric approximation in the mean is likewise closely related to integrated Lipschitz conditions; proofs of the theorems stated by Hardy and Littlewood were first published by Quade [l]. The object of the present note is to indicate rapidly and with a minimum of detail, that degree of approximation in the mean by polynomials in the complex variable and by bounded analytic functions is also conveniently investigated by use of integrated Lipschitz conditions. The investigation leads naturally to the use also of classes of analytic functions satisfying integrated Zygmund and integral asymptotic conditions. We shall approximate functions on an analytic Jordan curve or set of curves in the z-plane. Throughout the paper 7 denotes the unit circle, C a single analytic Jordan curve or a finite number X of mutually exterior analytic Jordan curves Cy, j=l, 2, • • , X; if X = l, Ci = C. A function z = Xi(w), w = reie, maps the closed interior of y: \w\ =1 onto the closed interior of Cy one-to-one and conformally; w = fl,(z) denotes the function inverse to z = Xy(-0; Cy denotes the interior of Cy together with Cy. Similarly, w = cp(z) maps the exterior of C conformally (not necessarily one-to-one) onto | w\ > 1 so that 00 = 1) is the image in the exterior of C ol \w\ =R under this mapping. In this paper, the letter s will denote arc-length measured on C, the letter p a number not less than unity, the letter k a non-negative integer unless otherwise indicated, the letters M and L with or without subscript constants independent of re, z and w. A function f(z) belongs to the class Hp on y if it is analytic interior to y and if Mp[f(rew)}={fln\f(reiB)Yd6}1'r> is bounded for 0

Sz.-Nagy's famous theorem states that a bounded operator $T$ which acts on a complex Hilbert space $\mathcal{H}$ is similar to a unitary operator if and only if $T$ is invertible and both $T$ and $T^{-1}$ are power bounded. There is an equivalent reformulation of that result which considers the self-adjoint iterates of $T$ and uses a Banach limit $L$. In this paper first we present a generalization of the necessity part in Sz.-Nagy's result concerning operators that are similar to normal operators. In the second part we provide characterization of all possible strong operator topology limits of the self-adjoint iterates of those contractions which are similar to unitary operators and act on a separable infinite-dimensional Hilbert space. This strengthens Sz.-Nagy's theorem for contractions., Comment: 13 pages, accepted for publication in Proceedings of the AMS

If X X is a compact subset of a Banach space with X − X X-X homogeneous (equivalently ‘doubling’ or with finite Assouad dimension), then X X can be embedded into some R n \mathbb {R}^n (with n n sufficiently large) using a linear map L L whose inverse is Lipschitz to within logarithmic corrections. More precisely, there exist c , α > 0 c,\alpha >0 such that \[ c ‖ x − y ‖ | log ⁡ ‖ x − y ‖ | α ≤ | L x − L y | ≤ c ‖ x − y ‖ for all x , y ∈ X , ‖ x − y ‖ > δ , c\ \frac {\|x-y\|}{|\,\log \|x-y\|\,|^\alpha }\le |Lx-Ly|\le c\|x-y\|\quad \mbox {for all}\quad x,y\in X,\ \|x-y\|>\delta , \] for some δ \delta sufficiently small. It is known that one must have α > 1 \alpha >1 in the case of a general Banach space and α > 1 / 2 \alpha >1/2 in the case of a Hilbert space. It is shown in this paper that these exponents can be achieved. While the argument in a general Banach space is relatively straightforward, the Hilbert space case relies on the fact that the maximum volume of a hyperplane slice of a k k -fold product of unit volume N N -balls is bounded independent of k k (this provides a ‘qualitative’ generalisation of a result on slices of the unit cube due to Hensley (Proc. AMS 73 (1979), 95–100) and Ball (Proc. AMS 97 (1986), 465–473)).

This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain $\Omega\subset \reals^n$, $n\geq2$, with boundary that is decomposed as $\partial\Omega=D\cup N$, $D$ and $N$ disjoint. We let $\Lambda$ denote the boundary of $D$ (relative to $\partial\Omega$) and impose conditions on the dimension and shape of $\Lambda$ and the sets $N$ and $D$. Under these geometric criteria, we show that there exists $p_0>1$ depending on the domain $\Omega$ such that for $p$ in the interval $(1,p_0)$, the mixed problem with Neumann data in the space $L^p(N)$ and Dirichlet data in the Sobolev space $W^ {1,p}(D) $ has a unique solution with the non-tangential maximal function of the gradient of the solution in $L^p(\partial\Omega)$. We also obtain results for $p=1$ when the Dirichlet and Neumann data comes from Hardy spaces, and a result when the boundary data comes from weighted Sobolev spaces., Comment: 36 pages

We discuss the dynamic and structural properties of polynomial semigroups, a natural extension of iteration theory to random (walk) dynamics, where the semigroup $G$ of complex polynomials (under the operation of composition of functions) is such that there exists a bounded set in the plane which contains any finite critical value of any map $g \in G$. In general, the Julia set of such a semigroup $G$ may be disconnected, and each Fatou component of such $G$ is either simply connected or doubly connected (\cite{Su01,Su9}). In this paper, we show that for any two distinct Fatou components of certain types (e.g., two doubly connected components of the Fatou set), the boundaries are separated by a Cantor set of quasicircles (with uniform dilatation) inside the Julia set of $G.$ Important in this theory is the understanding of various situations which can and cannot occur with respect to how the Julia sets of the maps $g \in G$ are distributed within the Julia set of the entire semigroup $G$. We give several results in this direction and show how such results are used to generate (semi) hyperbolic semigroups possessing this postcritically boundedness condition., To appear in Trans. Amer. Math. Soc

In this paper we present the following results: a description, via the Laplace transformation of analytic functionals, of the dual to the (DFS)-space A − ∞ ( D ) A^{-\infty }(D) ( D D being either a bounded C 2 C^2 -smooth convex domain in C N \mathbb {C}^N , with N > 1 N>1 , or a bounded convex domain in C \mathbb {C} ) as an (FS)-space A D − ∞ A^{-\infty }_D of entire functions satisfying a certain growth condition; an explicit construction of a countable sufficient set for A D − ∞ A^{-\infty }_D ; and a possibility of representating functions from A − ∞ ( D ) A^{-\infty }(D) in the form of Dirichlet series.