Search

In this paper, we shall generalize the unicellularity of operators on finite-dimensional spaces to that of the contraction of class Co on Hilbert spaces. We prove: (1) Each nilpotent operator on Hilbert space is Banach reducible (Theorem 3). (2) A contraction T of class CO on Hilbert space is unicellular if and only if T has one-point spectrum and every invariant subspace for T is cyclic (Theorem 6). (3) A contraction T of class Co on Hilbert space is unicellular if and only if T has one-point spectrum and all invariant subspaces of T are hyperinvariant subspaces of T (Theorem 8). The aim of this paper is to discuss the unicellularity of contractions of class Co. We shall generalize the unicellularity of operators acting on finitedimensional spaces to that on infinite-dimensional spaces. Throughout this paper, we denote by H a complex, separable, and infinitedimensional Hilbert space. The related concepts and symbols can be found in [4]. Definition 1. Let H1 be a nontrivial invariant subspace of T. H1 is called a Banach reducible subspace of T if there exists another nontrivial invariant subspace H2 of T such that

It is shown that on Lp[O, 1] all bounded linear operators which are Hermitian in the Calkin algebra B(LP)/C(4), must be of the form "Hermitian plus compact". That is, essentially Hermitian operators have the form, real multiplier plus compact. 1. Let X denote an infinite dimensional complex Banach space and B(X) the corresponding space of bounded (resp. compact) linear operators on X. The Calkin algebra associated with X is given by A(X) = B(X)/C(X). Many papers recently have dealt with variations of the following lifting question: Given that a coset T + C(X) in A (X) has a certain property, does the coset "lift" to an operator T + K, K E C(X), having the same property? For example, Stampfli [8] has shown that, if X is a separable complex Hilbert space, for every operator T E B(X) there is a compact operator KT so that the Weyl spectrum of T and the spectrum of T + KT are equal. In fact for most lifting theorems X is a separable infinite dimensional complex Hilbert space. Recently however, in an attempt to consider more general Banach spaces, these authors have proved that if X = Ip, then Hermitian elements in the Calkin algebra lift to the form "Hermitian plus compact". In this paper the above result is extended to the case X = Lp[O, 1] (hereafter referred to as L.): namely, the essentially Hermitian operators on B(Lp), 1 1, 4' is unique.) So given A,, 4'(t) = sgn 4'(t) I4(t) IP where sgn it = e-is if it = pei9. Finally, if 4 and 4' are unit vectors in L. then 4;8'(Qk>) = = f (t)4'(t) dt. We begin with a result proved in [1, Lemma 1], for T E B(Ql). The proof carries over to B(4L) with only one minor change. Whereas in the proof of [1] there existed a projection P E 6Y and unit vectors + and 4' in Lp for which = supp e q PTP-' , due to the fact that T and either P or P' were compact operators, here, it may only be assumed that for given E > 0, there exists a This content downloaded from 157.55.39.205 on Sun, 04 Dec 2016 04:55:31 UTC All use subject to http://about.jstor.org/terms ESSENTIALLY HERMITIAN OPERATORS 73 projection P E 'P and unit vectors 4 and 4' satisfying + > supp pE IPTP'Ilp. Nevertheless the following result still holds. LEMMA 1. Let T E B(LP), 1 < p < xo,p #'2. Then sup IIPTP'll < cpri(T) < so

Given a nonnegative Cl-function p(x) on a Riemannian manifold R, denote by BO(R) the Banach space of all bounded C2-solutions of Au = pu with the sup-norm. The purpose of this paper is to give a unified treatment of BO(R) on the Wiener compactification for all densities p(x). This approach not only generalizes classical results in the harmonic case (p 0), but it also enables one, for example, to easily compare the Banach space structure of the spaces BO(R) for various densities p(x). Typically, let ,8(p) be the set of all p-potential nondensity points in the Wiener harmonic boundary A, and Cp(A) the space of bounded continuous functions f on A with ft A 8(p) 0. Theorem. The spaces Bp(R) and Cp(A) are isometrically isomorphic with respect to the sup-norm. Throughout this paper R is an orientable Riemannian C?-manifold of dim > 2, and p(x) is a nonnegative Cl-function on R. Denote by Bp(R) the space of bounded C -solutions u on R of the elliptic equation Au = pu, where Au is the Laplacian of u on R. As one studies bounded harmonic functions on the Wiener compactification, the space B p(R) has been investigated on the so-called Wiener p-compactification (cf. Loeb and Walsh [2], Wang [91). However, their consideration restricts one to construct different compactifications for different densities p(x). The purpose of the present paper is to give a unified treatment of the spaces B (R) on the Wiener compactification R * for all densities p(x). p This approach, for instance, enables one to easily compare the linear space structure of the spaces B p(R) for various densities p(x). Typically, let ,8(p) be the set of p-potential nondensity points x in the Wiener harmonic boundary A (see below for its definition), and C p(A) the space of bounded Received by the editors May 31, 1973 and, in revised form, October 24, 1973. AMS (MOS) subject classifications (1970). Primary 30A48.

Let A be a ring with identity. It is well know that a group ring A[G] with a finite group G is Quasi-Frobenius (QF) iff A is QF. Using the concept of Frobenius extensions introduced by F. Kasch [4], we shall obtain a similar result for QF-3 rings in this paper, namely, A[ G] is left QF-3 iff A is left QF-3. Here a ring is called left QF-3 if it has a minimal faithful left module, that is, a faithful left module which is isomorphic to a direct summand of every faithful left module. Further we shall show that in case A is a G-Galois extension of the fixed subring A G relative to a finite group G of ring automorphism of A in the sense of [7], A is left QF-3 iff A G is left QF-3. It should be noted that A and A G are not always left QF-3 even if A G and A are left QF-3, respectively and that in case A /A G is finite G-Galois, A is QF whenever A G iS QF but the converse is not necessarily true. Throughout this paper, all rings, all modules, all subrings and all ring homomorphisms are assumed to be unitary. We follow the notation of [10] unless specified otherwise. For A -A'-bimodules A MA" A NA, the notation A MA IA NA' denotes the fact that AMA' is isomorphic to a direct summand of a direct sum N(') of finitely many copies of ANA,. A module M is said to be cofinitely generated (co-f.g.) in case for every set { Mi; i E I } of submodules of M if the intersection fl ,Mi = 0, then there exist il,... , in in I such that Mn kMi = 0 (see [11]). If a module M is f.g. projective, co-f.g. injective and faithful, then M will be called a *-module for convenience. If a ring A is left QF-3, then a minimal faithful left A-module is a *-module, and conversely if A has a left *-module, then A is left QF-3 (see [2, Theorem 1]). Let A iD B 3 1A be a ring extension. We say A is a Frobenius (resp. a left QF) extension of B if BA is f.g. projective and if AAB -(resp. j) AHom(BA, BB)B, and a right QF extension is defined symmetrically (see [4] and [8]). If A is a Frobenius extension of B, then there exist a B-B-homomorphism h of A to B and r1, . .. , r"; 11, . . ., In in A such that x = E:2rih(lix) = Eih(xri)li for all x in A, and conversely (see [9]). When this is the case, we shall call such a system (h; li, ri)in a Frobenius system.

A compact, orientable, Riemannian manifold of dimension n is considered, with the Laplace operator acting on the exterior 2-forms of the manifold. Examining the spectrum, Sp 2 {\text {Sp}^2} , of the Laplace operator acting on 2-forms, the question is raised whether Sp 2 {\text {Sp}^2} exerts an influence on the geometry of the Riemannian manifold. To answer this question, after some preliminaries, two compact, orientable, equispectral, i.e., having the same Sp 2 {\text {Sp}^2} , Riemannian manifolds are considered in §3. (We note, in particular, that equispectral implies that the two manifolds are equidimensional.) Assuming further that the second Riemannian manifold has constant sectional curvature, the paper exhibits all the dimensions, commencing with 2, for which the two Riemannian equispectral manifolds have the same constant sectional curvature. In particular, this implies that for certain dimensions, which are explicitly stated, the Euclidean n-sphere is completely characterized by the spectrum, Sp 2 {\text {Sp}^2} , of the Laplacian on exterior 2-forms. Next, two compact, orientable, equispectral, Einsteinian manifolds are considered. (Again, equispectral implies equidimensional.) Assuming that the second Einsteinian manifold is of constant sectional curvature, the paper exhibits all the dimensions for which the two Einsteinian equispectral manifolds have equal constant sectional curvature. In particular, taking the second manifold to be the standard Euclidean sphere, the paper classifies Einsteinian manifolds, which are equispectral to the sphere, by calculating all the dimensions for which the Einsteinian manifold is isometric to the sphere. In short, if one of the Einsteinian manifolds is the sphere, then for certain dimensions, equispectral implies isometric. In §4, compact, equispectral, Kählerian manifolds are considered, and additional conditions are examined which determine their geometry. Studying two compact, equispectral, Kählerian manifolds, and again assuming that one of the manifolds is of real, constant, holomorphic, sectional curvature, the paper exhibits all the dimensions for which the two manifolds have equal real, constant, holomorphic, sectional curvatures. As a particular case, the paper classifies all the dimensions for which complex projective space, with Fubini-Study metric, is completely characterized by the spectrum, Sp 2 {\text {Sp}^2} , of the Laplacian acting on exterior 2-forms. The calculations were performed by utilizing an electronic computer.

Let Kn be the set of n-positive maps of B(H) to B(H). A Knpositive map of a C*-algebra A to B(H) is a positive linear map q5 such that ETr(q5(ai)bt) > Oforany Ea,bi E {x E A0@T(H)jK n Dv a, (id0oa)(x) > 0}. It is shown that the following three statements are equivalent. (1) Every K -positive map of A to B(H) is Kf+'-positive. (2) Every Kn-positive map of A to B(H) is completely positive. (3) A is an n-subhomogeneous C*-algebra. Introduction. The concepts of mapping cones K and K-positive maps which have introduced by St0rmer in [4] are powerful tools when one considers the extension problem of positive maps in C*-algebras. In this paper, we investigate Kn-positive maps of a C*-algebra A to the algebra B(H) of all bounded operators on a Hilbert space H, which are induced by the intermediary of the set of all n-positive maps Kn (which is one of the mapping cones) in B2 (H)+, and which may have a close relation with the extension problem. The positivity of Kn-positive maps is stronger than the positivity of B2 (H)+positive maps and weaker than the positivity of CP(H)-positive maps, and becomes stricter as n grows large. For the definitions of mapping cones, B2 (H)+, and CP(H) we refer the reader to the paper [4]. According to Proposition 1 in the next section, a Kn-positive map is an npositive map. This is not true, however, in the reverse direction, as it is known that a 1-positive map is not in general K1-positive (that is B2 (H)+-positive). The author has recently shown in [2] that the cone which corresponds to the n-positive maps of A to B(H) is n=onv-Y{ ( ai 0b ) (Z, ai 0 bi) a. E Al bi T(H), i = 1 n} in (A0,,T(H))+, where T(H) is the set of all trace class operators on H. Combined with the results of [4], the relation among the cones in (A&,,T(H))+ is the following: cA c ... c cA C CA C ... C (A T (H))+ n n n 1 P(A, K') c ... c P(A, Kn) c P(A, Kn+1) c ... c P(A, CP(H)) When we study the extension problem of positive maps in C*-algebras, it is important whether the above inclusions are strict or not. Thus, we encounter a problem which is similar to the conjecture posed by M. D. Choi [1] in 1972. He Received by the editors January 21, 1986 and, in revised form, May 21, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46L05.

This paper deals with certain properties of the subspaces of LC(H) and Cp, and namely those connected with the reflexivity and with the property of containing classical spaces. It is proved that any subspace of Cp (1 < p < o) is either isomorphic to Hilbert space or it contains a subspace isomorphic to Ip. For C1 and LC(H) the same results were obtained by J. R. Holub, cf. [4]. Introduction. Let H be a complex Hilbert space. Denote by LC(H) or C., the space of all compact operators on H, with the usual operator norm. Let Cp (1 < p < oo) be the space of all operators T: H -_ H for which ITIP = trace(T*T)P/2 < o with the norm |. IP LC(H) and Cp are Banach spaces. The word "subspace" in this paper will mean "closed subspace". For the subspaces of LC(H) and Cp it is known that [1] if X is a subspace of LC(H) which is complemented in the space L(H) of all bounded operators on H, then X is a reflexive space. J. R. Holub [4] proved that if a subspace X of LC(H) or C1 is isomorphic to H, then it is complemented in L(H), and if X is not isomorphic to H it contains a subspace isomorphic to co or 1 respectively. We will prove results similar to those of Holub for the Cp spaces (1 < p < oo). We shall denote an orthogonal projection from H onto a subspace E by pEE For any two subspaces E, F of H and any 1 < p < oo, REF will denote the subspace of Cp: REF= KA c Cp: (1 PF)A(1PF.) o1. This subspace is closed in the norm | Ip. Proposition 1. If E and F are finite dimensional subspaces of H, then R EF is isomorphic to a Hilbert space for any 1< p < . p Proof~. It is easily seen that REF C REF. Presented to the Society, July 30, 1973; received by the editors July 22, 1974 and, in revised form, September 18, 1974. AMS (MOS) subject classifications (1970). Primary 47B05, 47B10; Secondary 46B 10.

Introduction. In two previous notes [9],1 I stated some results which, in a general way, may be expressed as follows: If the Taylor series which represents an entire function satisfies a certain gap condition (which depends only on the order of F(z)), the zeros of F(z) -f(z) are not exceptional with respect to the proximate order of F(z) by any meromorphic function f(z) 3 oX of lower order. On the other hand, Mandelbrojt (see for instance [3, Theorem XXV]) proves that an entire function represented by a Dirichlet series takes each value a 0 oc, except at most one, in any horizontal strip of width greater than a quantity which depends only on the order (R) and on the upper density of the sequence of exponents of the series. In the present note we shall prove some results closely related to those of my above mentioned notes and to that of Mandelbrojt just quoted. These results may briefly be stated as follows: If an entire function F(s) can be represented by a Dirichlet series satisfying certain gap conditions, the zeros of F(s) -a cannot be exceptional with respect to the proximate order (R) of F(s) in any strip of width greater than a quantity, determined by the order, for every value of a $ oX without exception. In ?1 we shall deal with functions of finite order (R); in ?2, we shall give results concerning functions of infinite order (R). I think it will be of interest to remark that the theorems of the type of the well known Hadamard's theorem (which asserts that a Taylor series satisfying a specific gap condition cannot be continued analytically outside its circle of convergence) together with the results concerning relationship between gap properties and the position of the Julia lines, and again together with the results of my above mentioned notes and those contained in one of my papers [10], enable us to state the following general principle (without pretending that it holds for every case). By means of gap conditions in the Taylor series the disappearance of the possibility of the existence of the exceptional cases can be affirmed. Therefore, the content of this paper may be regarded as a link

This paper contains a proof that either all, or none, of the nontrivial solutions of the fourth order linear selfadjoint differential equation have an infinite number of zeros on a half line, provided that no nontrivial solution has more than one double zero on that half line. Throughout this paper, let Ly = (ry")" -(qy')'+py where r, q, and p are given real-valued functions, a?(o, oo) is given, r", q', pCC[a, oo), and r(t)>O for t_?a. A nontrivial solution to Ly=O is said to oscillate if its zeros in [a, oo) are unbounded. THEOREM 1. If no nontrivial solution to Ly = 0 has more than one double zero in [a, oo), then all the nontrivial solutions oscillate or none oscillate. W. Leighton and Z. Nehari [1, p. 367 ] obtain the same conclusion using the hypothesis that q(t) 0, p(t) > 0 for t > a. As they note, these assumptions imply that no nontrivial solution has more than one double zero. Some lemmas will be established before proving Theorem 1. Ly = 0 will be said to be 2-2 disconj ugate if no nontrivial solution has more than one double zero in [a, oo). Of course, this is a special case of the concept known as n-n disconjugacy. When r, q, and p are all constants, it can easily be shown that Ly =0 is 2-2 disconjugate if and only if rw4+qw2+p > 0 for all real numbers, w. For i = 1, 2, 3 and a < b < oo, let ybi designate the solution to Ly = 0, y(i)(b) = bij, j = 0, 1, 2, 3, where 6ij is the Kronecker delta. Denote the zeros of Yb3 by ,[ . . < -q(b .1 < nb 1)

The purpose of this paper is to give a necessary and also a sufficient condition for a Dirichlet finite harmonic function on a Riemann surface to be represented as a difference of a Dirichlet finite solution of Au=Pu (P_O) and a Dirichlet finite potential of signed measure. 1. Let P=P(z) dx dy (z=x+iy) be a nonnegative not identically zero o-Holder continuous (0<1) second order differential on a Riemann surface R and PD(R) be the Hilbert space of all Dirichlet finite solutions of (1) AU(Z) = P(Z)U(Z), A = 4a2./I aZa, on R with the scalar product given by mixed Dirichlet integral, i.e. (u, v)=D11(u, v)=fS11 duA*dv, not the energy integral. The study of PD(R) was begun by Royden [6]. We will use the fact shown by Nakai [2] that PD(R) forms a vector lattice under the natural order in PD(R). We also use the Glasner-Katz maximum principle [1] that the modulus of every function in PD(R) takes its maximum on the Royden harmonic boundary. The recent result of Nakai [3] that PBD(R) is dense in PD(R) will not be made use of. Let A(R) be the Royden harmonic boundary and HD(R) be the class of Dirichlet finite harmonic functions on R. (For the basic materials from the Royden compactification and the class HD(R) we refer to the monograph of Sario and Nakai [7].) One of the important problems in the theory of PD(R) which is not fully developed yet is to describe the distribution of PD(R)|A(R) in HD(R)|A(R). We will prove a theorem which contributes to this question. 2. If R is parabolic, then PD(R)= {0} (cf. Royden [6]), which case offers no interest. Therefore we assume throughout the paper that R is hyperbolic. Let M(R) be the class of all Dirichlet finite Tonelli functions on R and MA(R) the subclass of M(R) consisting of functions f with f JA(R)=O (cf. [7]). We then have the orthogonal decomposition M(R) = HD(R) + MA(R), Received by the editors May 26, 1971. AMS 1970 subject classifications. Primary 31A05, 58G99. 1 The work was sponsored by the U.S. Army Research Office-Durham, Grant DA-ARO-D-31-124-71-G20, UCLA. (e, American Mathematical Society 1972

In this paper we show that if a smoothly bounded, relatively compact domain in a complex manifold admits a complete Kähler metric with certain bounds on its Ricci tensor, then the domain must be holomorphically convex. This gives an obstruction for the existence of a complete Kähler-Einstein metric on such domains.

Let (Mz1, ... , MZn ) be an n-tuple of shift operators on the polydisk 12(Zn); we compress it to a variety of subspaces of 12(Zn) that are combinatorially constructed. The main result is a multivariate Fredholm index formula, which links the indices of the n-tuples to their combinatorial data in the definitions of the subspaces. In [11], Taylor, using homological algebraic methods, introduced a natural spectral theory for n-tuples of commuting operators acting on Banach spaces. Fredholm theory and index theory are developed with respect to this spectral theory [2, 3, 8]. While this several variable spectral theory is a natural generalization of the classical single operator theory (when n = 1), the techniques of computing these spectra are so different from or, much more complicated than, that of single operators. The problem of deciding the spectral pictures of various classes of n-tuples of operators is very essential in multivariate operator theory and has deep applications to many areas of analysis, such as function theory on domains in Cn . Many examples have been investigated through many different techniques [4, 5, 7, 9, 10]. In this note, we use the homological method to compute the spectra and Fredholm indices of a new, interesting class of n-tuples of operators. We will give an index formula for these n-tuples in terms of combinatorial data in their definition. An interesting consequence of this is that some combinatorial property of the operator forces the index to be nonzero. To be more precise, we denote by T = (T1, . .. , Tn) an n-tuple of commuting operators on the Hilbert space H and by K(T, z) = {Kq (T, z), 0q } the Koszul complex of T at z E Cn, where Kq (T, z) is the space of q-cochains and aq is the differential map at degree q. We refer the reader to Taylor's original papers [1 1, 12] for a more detailed account of these concepts. Recall that a point z in Cn is said to be in the Taylor spectrum a(T), if the Koszul Received by the editors October 6, 1992. 1991 Mathematics Subject Classification. Primary 47D25. Research supported in part by a grant from NSF, DMS 90-02969. (? 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page

For a countable set Ω ⊂ R n \Omega \subset {\mathbb {R}^n} denote by P ( Ω ) P(\Omega ) the space of polynomials spanned by x ω , ω ∈ Ω ( x = ( x 1 , … , x n ) ∈ R n , ω = ( ω 1 , … , ω n ) ∈ Ω , x ω = ∏ i = 1 n x i ω i ) {x^\omega }, \omega \in \Omega (x = ({x_1}, \ldots ,{x_n}) \in {\mathbb {R}^n}, \omega = ({\omega _1}, \ldots ,{\omega _n}) \in \Omega , {x^\omega } = \prod _{i = 1}^nx_i^{{\omega _i}}) . In this paper we investigate the question of the density of P ( Ω ) P(\Omega ) in C ( K ) C(K) , the space of real valued continuous functions endowed with the supremum norm on compact set K ⊂ R n K \subset {\mathbb {R}^n} . In case n = 1 n = 1 the classical theorem of Müntz gives an elegant necessary and sufficient condition for density. This problem (closely related to the distribution of zeros of Fourier transforms) is much more complex in the multivariate setting. We shall present an extension of Müntz’ condition to the case n > 1 n > 1 which will suffice for density. This, in particular, will enable us to construct "optimally sparse" lattice point sets Ω \Omega for which density holds.

This paper gives a proof of the theorem that for Coxeter groups the algebra of coinvariants is isomorphic to the normalizer of the Coxeter group in the linear group of a vector space over R. I have tried to give a relatively elementary proof which requires only elementary algebra and a little knowledge of the theory of invariants

In this paper some new integral inequalities of the Poincaré type, on a region of rectangular dimensions, involving many functions in many variables are obtained. These in turn can be used to serve as generators of other integral inequalities.

In this paper, we obtain necessary and sufficient conditions for a co-semigroup with continuity in the uniform operator topology for t > 0 in Hilbert space in terms of the spectral property of its infinitesimal generator. Then we get the characteristic theorem for the compact semigroup.

Let H {\mathbf {H}} be a Hilbert space and f : H → H f:{\mathbf {H}} \to {\mathbf {H}} a continuous map which is expanding (i.e., | | f ( x ) − f ( y ) | | ≥ | | x − y | | ||f({\mathbf {x}}) - f({\mathbf {y}})|| \geq ||{\mathbf {x}} - {\mathbf {y}}|| for all x , y ∈ H {\mathbf {x}},{\mathbf {y}} \in {\mathbf {H}} ) and such that f ( H ) f({\mathbf {H}}) has nonempty interior. Are these conditions sufficient to ensure that f f is onto? This question was stated by Nirenberg in 1974. In this paper we give a partial negative answer to this problem; namely, we present an example of a map F : H → H F:{\mathbf {H}} \to {\mathbf {H}} which is not onto, continuous, F ( H ) F({\mathbf {H}}) has nonempty interior, and for every x , y ∈ H {\mathbf {x}},{\mathbf {y}} \in {\mathbf {H}} there is n 0 ∈ N {n_0} \in \mathbb {N} (depending on x {\mathbf {x}} and y {\mathbf {y}} ) such that for every n ≥ n 0 n \geq {n_0} \[ | | F n ( x ) − F n ( y ) | | ≥ c n − m | | x − y | | ||{F^n}({\mathbf {x}}) - {F^n}({\mathbf {y}})|| \geq {c^{n - m}}||{\mathbf {x}} - {\mathbf {y}}|| \] where F n {F^n} is the n n th iterate of the map F , c F,c is a constant greater than 2, and m m is an integer depending on x {\mathbf {x}} and y {\mathbf {y}} . Our example satisfies | | F ( x ) | | = c | | x | | ||F({\mathbf {x}})|| = c||{\mathbf {x}}|| for all x ∈ H {\mathbf {x}} \in {\mathbf {H}} . We show that no map with the above properties exists in the finite-dimensional case.

We have obtained two main results by using probabilistic methods : (i) For a domain, we obtained a representation formula of the bounded solution to the first boundary value problem for Schrodinger equation; (ii) For α∈R 1 , under certain conditions, we proved that the bounded solution having limit α at infinity to the generalized first boundary value problem for Schrodinger equation exists and is unique, and it is represented in explicit formula. The results of this paper are generalizations of Chung and Rao

For a three-dimensional model of a positive feedback loop, Selgrade [11, 12] proved that every positive-time trajectory in the nonnegative orthant converges. Hirsch [6] gave another proof of this result under slightly different assumptions. This paper provides a new proof of Selgrade's result that is much shorter and presents a generalization that can be applied to positive and negative feedback loops and other systems.

This note shows that the Davenport-Hasse relation for Gauss sums is equivalent to the evaluation of some multidimensional exponential sums that generalize that of Salie. This paper shows that the Davenport-Hasse relation for Gauss sums is equivalent to the evaluation of some multidimensional exponential sums that generalize that of Salie. In addition to being interesting in their own right, these sums also occur in the analysis of Fourier expansions of certain Poincare series on the zj-fold cover of GLin). For example, if one considers the analogues of the exponential sums in Theorem 1 of M. Larsen's appendix to [B-F-G] that occur for the Poincare series on the 3-fold cover of GL(3), the hyper-Kloosterman sums that arise are of the type considered here when n = 3 . Thus the evaluation given below may turn out to be important in the further development of the theory of such automorphic forms when zz > 3. This is certainly the case for n = 2, as the work of Iwaniec [I] shows. Let F = GF(q), a = pr for p prime, and n £ Z+ be such that q = 1 (mod«). For \p a multiplicative character of order n and a £ F* consider the n -dimensional Salie sum defined by Snia)= ^2 ¥(x2x3---xn {) eq(xx + ■ ■ ■ + xn) , x,x2-x„=a Xi£F* where eq(x) = e(trx/p). (Note that Sn(a) does not depend on the choice of ¥■) Theorem. (*) Sn(a) = eq,nq("-xV2 £ eq(nx)

This paper completes an investigation of «jumps» of orderings. The last few cases are given in the proof that for each recursive ordinal α≥1 and for each Turing degree d≥0 (α) , there is a linear ordering A such that d is least among the αth jumps of degrees of (open diagrams of) isomorphic copies of A, and for β

The paper is concerned with the growth properties at infinity of the propagators C(.), S(.) of the equation u″(t)+Bu'(t)+Au(t)=0, Where A, B are densely defined closed linear operators in a Hilbert space. We define ω 0 (A,B)=max{lim t→ ∞t −1 lnS'(t)}, and give a criterion to judge whether ω 0 (A,B)≤b for a fixed b∈R

We construct a cellular wedge A V B c R3 such that A is not cellular. In a recent list of problems circulated by R. J. Daverman, the following question [1, El 9] is attributed to Morton Brown: If a wedge A V B c R3 is cellular, is A cellular? (Daverman was able to construct cellular wedges in R n, n > 5, having a noncellular constituent. His examples are included in an addendum to this paper.) The following example answers this question negatively. Example. Let A c R3 be the well-known Fox-Artin arc [5, Example 1.3], constructed so as to be locally polyhedral modulo its endpoints p and q. There is a piecewise linear 3-cell whose interior contains p and whose boundary is pierced by A at a single point. Guided by A, we "bore out" a tunnel in the 3-cell pinching down to p, and so construct a crumpled cube B such that B is not a 3-cell, Bd B is locally polyhedral modulo p, and A n B = {p}; the process of obtaining B is more precisely described in [11]. The arc A is not cellular because, as shown explicitly in [5, Example 1.3], R A is not an open 3-cell (in particular, Fox and Artin show that there is a simple closed curve in R3 -A that is not contained in the interior of a 3-cell in R3 A). On the other hand, X = A U B is cellular. The argument for this is well known, but we give a brief sketch here for completeness. To begin, note that there are arbitrarily small 3-cells C such that B n C is a disk whose interior contains p and B U C is a piecewise linear 3-cell whose boundary is pierced by A at a single point. It is easily seen that Y = X U C is cellular by constructing a 3-cell neighborhood of Y in the e-neighborhood of Y. To accomplish this, begin with a small piecewise linear sphere Y about q that is disjoint from B and is pierced by A at three points. Let these points be denoted by al, a2 , and a3, ordered moving from p to q. Let D1, D2, and D3 be pairwise disjoint small disks on X about a1, a2, and a3, respectively. Within the e-neighborhood of Y, stretch D1 in Received by the editors May 15, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 57N60.

In this paper we calculate the contribution to the trace formula of those orbital integrals coming from matrices in SL(3, Z) with two complex eigenvalues and one real one, none of which are equal to zero. These correspond to mixed cubic number fields and will be seen to occur with multiplicity equal to the class number of a certain order in the number field

Let u ( x , t ) u(x,t) be the solution of the Schrödinger equation with initial data f f in the Sobolev space H − 1 + a / 2 ( R n ) {H^{ - 1 + a/2}}({\mathbb {R}^n}) with a > 1 a > 1 . This paper shows that the weighted inequality ∫ R n ∫ R | u ( x , t ) | 2 d t ( 1 + | x | ) − a d x ≤ C ‖ f ‖ H − 1 + a / 2 ( R n ) \int _{{\mathbb {R}^n}} {\int _\mathbb {R} {{{\left | {u(x,t)} \right |}^2}dt{{(1 + \left | x \right |)}^{ - a}}dx \leq C{{\left \| f \right \|}_{{H^{ - 1 + a/2}}({\mathbb {R}^n})}}} } is false. Another improved weighted inequality is proved for the general case.

In the present paper, we have discussed the number of power roots of linearized polynomials. For some cases, the exact formulas are given

Let R R be a ring with center Z ( R ) Z(R) . A mapping F F of R R into itself is called centralizing if F ( x ) x − x F ( x ) ∈ Z ( R ) F(x)x - xF(x) \in Z(R) for all x ∈ R x \in R . The main result of this paper states that every additive centralizing mapping F F on a von Neumann algebra R R is of the form F ( x ) = c x + ζ ( x ) , x ∈ R F(x) = cx + \zeta (x),x \in R , where c ∈ Z ( R ) c \in Z(R) and ζ \zeta is an additive mapping from R R into Z ( R ) Z(R) . We also consider α \alpha -derivations and some related mappings, which are centralizing on rings and Banach algebras

The spans of simple closed curves that are boundaries of convex regions are determined. The following question of Lelek is answered affirmatively for these curves: Are the span and the semispan of a simple closed curve equal? The same answer is obtained in the case of nonconvex quadrilaterals. INTRODUCTION The concept of the span of a metric space X was introduced by A. Lelek [1]. In [2], he defined three other versions of span called surjective span, semispan, and surjective semispan. In this paper we concern ourselves with the span of the boundaries of convex regions in the plane. We also deal with the simple case of a nonconvex polygon. We begin by recalling the definitions. Let X be a connected nonempty metric space. The span a(X) of X is the least upper bound of the set of real numbers r > 0 satisfying the following condition: There exists a connected space Y and a pair of continuous functions f, g: Y -X such that f(Y) = g(Y) and dist[f(y), g(y)] > r for y E Y. The surjective span v*(X) of X is defined similarly. The only difference is that we impose one additional condition on the mappings f and g, namely, f(Y) = X = g(Y). To obtain the definition of the semispan ao(X) of X we replace the condition f(Y) = g(Y) in the definition of v(X) by the inclusion f(Y) c g(Y), and to define the surjective semispan a*(X) of X we additionally require that g(Y) = X (while f(Y) c g(Y)). The following inequalities were established in [2]: (1) 0 < a* (X) < 5(X) < ?o(X) < diamX, (2) 0 < u*(X) < a*(X) < ao(X) < diamX for every connected nonempty metric space X. We now introduce the notion of a directional diameter. Let X be a simple closed curve in the Cartesian plane. Let L. denote the line passing through Received by the editors February 20, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 54F20. The results in this papor were announced in the Abstracts of the AMS, October 1988, Issue 59, Volume 9, Number 5, p. 425. ? 1991 American Mathematical Society 0002-9939/91 $1.00 + $.25 perpage

In this paper we prove that if n > 1 and 0 < p < 1 then the Lumer's Hardy space (LH)p(Bn) of the unit ball Bn in Cn does not have the Orlicz-Pettis property.

We give an example of a noncompact, unimodular group G with the property that B(G) n Co(G) = A(G), where A(G) is the Fourier algebra of G, B (G) is the Fourier-Stieltjes algebra of G and Co(G) is the set of all complex, continuous functions on G vanishing at infinity. This example answers negatively a question raised by A. Figa-Talamanca. 1. In a paper entitled Positive definite functions which vanish at infinity Alessandro Figa-Talamanca constructs a "singular" continuous, positive defi- nite function, i.e., one not in the Fourier algebra of G, which vanishes at infinity for any locally compact unimodular group that satisfies the condi- tion: (H) The von Neumann algebra, M (X), generated by the left regular repre- sentation A of G is not purely atomic. Groups satisfying (H) are of necessity not compact. In the aforementioned preprint, A.Figa-Talamanca notes that it is unknown whether or not unimod- ularity alone is sufficient (for a noncompact, locally compact group) to guarantee the existence of a "singular" continuous, positive definite function which vanishes at infinity. We show by means of an example in the following section that unimodularity alone is not sufficient to guarantee the existence of a positive definite function with the aforementioned properties. 2. In the fourth section of (3) the following group and its representation theory are discussed. We will briefly discuss this example with sufficiently many references so that the interested reader can fill in the details. Namely

This paper is concerned with characterising, in terms of certain properties of their Fourier transforms, the Lipschitz functions of order α ( 0 > α > 1 ) \alpha (0 > \alpha > 1) defined on a locally compact metric 0 0 -dimensional Abelian group.

It is shown that there exists f E H1(U3) which is not the product of two functions in H2(U3). This partially answers a question posed by W. Rudin. We let U be the unit disc in the complex plane, T the unit circle, Un and Tn the Cartesian products of n copies of U and T respectively, m normalized Lebesgue measure on Tn, and HP(Un) the usual Hardy class of holomorphic functions on Un. It is well known that every f E H'(U) can be expressed as the product of functions g and h in H2(U). In fact if B denotes the Blaschke product of the zeros of f, it is sufficient to let g = f/B and h = Bg. Rudin [1], [2, p. 631 has shown that if n > 4, then not every f E Hl(Un) can be factored into a product of functions in H2(Un). It is the purpose of this note to show that Rudin's method may be extended to cover the three-dimensional case as well. Theorem. There exists f E HW(U3) which is not expressible as a product of two functions in H2(U3). Thus we observe that for this factorization problem only the case n = 2 remains open. That case does not seem to be accessible by these techniques. (See the last paragraph of this paper.) We begin with two lemmas. Lemma 1. The ratio jl(z + w)2,11 1/jj(z + w)2qjj 2 tends to zero as the integer q tends to infinity.

We investigate definable automorphisms of ^"(uj/fin and show that e.g. every Borel automorphism is trivial. The existence of nontrivial projective automor- phisms is consistent and independent from ZFC + CH. Throughout this paper a2(w)/fin will denote the Boolean algebra of 0>(to) modulo the ideal of finite sets. Call an automorphism F of ^(c^/fin trivial if there are two cofinite sets a,b e u and a bijection /: a -> b such that F(x) = (f"(a n x)). Here (x) denotes the equivalence class of x under the equivalence relation x =+ y iff (x\y)U (y\x) is finite. Note that there are exactly 2K° trivial automorphisms. It is a well-known fact, first proved by W. Rudin (7), that the Continuum Hypothesis implies there are 22 ° automorphisms of £P(o:)/nn, hence most of them are nontrivial. It is also consistent with —,CH that there is a nontrivial automorphism of aa(w)/fin. This was proved by Baumgartner (unpublished). He noticed that if there is a family {aa: a < ux) of subsets of w such that (i) aa \ as is finite for every a < s < «,, (ii) {aa: a

Let SIR be a ring extension, where S is a commutative ring. If SIR is strongly separable then it can be imbedded in a weakly Galois extension of R in the sense of [7, Definition 3.1]. Throughout this paper, all rings will be assumed commutative. Moreover, R will mean a ring with identity element 1, and all ring extensions of R will be assumed with identity element coinciding with the identity element of R. A ring extension SIR is called strongly separable if S is a separable R-algebra which is projective as an R-module (and so, S is a finitely generated R-module by [6, Villamayor's theorem]). Moreover, a ring S will be called connected if S has no proper idempotents. In [1], Auslander and Goldman proved that if SIR is a ring extension which is strongly separable and S is a free R-module then SIR is imbedded in a Galois extension of R. Moreover, in [4], Janusz proved that if SIR is a ring extension which is strongly separable and S is connected then SIR is imbedded in a Galois extension N of R such that N is connected. Now, in [7], Villamayor and Zelinsky gave the notion of weakly Galois extensions. A ring extension NIR is weakly Galois if and only if there exists a finite number of orthogonal idempotents el, *, et in R such that =lei= 1 and the extensions Nei/Rei are Galois ([7, 3.15]). In [5], Magid proved that some types of rings have separable closures, and strongly separable extensions of the rings are determined by the Galois groupoids of the closures. In this note, we shall prove the following theorem by using the results of [7]. THEOREM. Every ring extension SIR which is strongly separable can be imbedded in a weakly Galois extension of R. To prove the theorem we need several lemmas. Now, let R be a connected ring, and S/R a ring extension which is strongly separable. Then Received by the editors March 23, 1973. AMS (MOS) subject classifications (1970). Primary 13B05.

R. C. Reilly calculated the variations of functions of the mean curvatures for hypersurfaces in Euclidean space. In the present paper, using Reilly’s formulas, we derive some general integral formulas for hypersurfaces, which generalize the well-known Minkowski formulas, and then apply those formulas to obtain some characterizations of the hypersphere.

In this paper a theorem on the absolute summability of an integral associated with the allied integral of a Fourier integral by a functional Norlund method has been established. Our theorem generalises an earlier result due to Boicun on the absolute Norlund summability of the allied integral. 1. Let p(t) be a real locally integrable function on the real half-line. If f is locally integrable in (0, oo), P(y) =j'p(t) dt, and (y) yP(Y u)f(u) du, then the integral fo f (u) du is said to be absolutely (N, p(y)) summable or summable IN, p(y)I if fO Ia'(y)I dy O) the method of summability IN,p(y)l is the same as the well-known Cesaro's method IC, al. 2. Letf(t) E L(xc, so). The allied integral of a Fourier integral of the function f (t) is defined [3] (2.1) ? duf f(t)sin(t x)u dt. 'I o 00 The absolute N6rlund summability of Fourier series and its allied series has been studied by several workers. The corresponding problem of absolute summability of trigonometric integrals by the functional N6rlund Received by the editors September 22, 1972 and, in revised form, February 26, 1973 and March 29, 1973. AMS (MOS) subject classifications (1970). Primary 40G05; Secondary 42A92.

Let T be a bounded linear operator from Li(G, A) into Li(f!,7, P), where (G, A) is a compact abelian metric group with its Haar measure, and (f2, 7, P) is a probability space. Let (sw) be the random mea- sure on G associated to T; that is, T/(oj) = fG f(t) dpu(t) for each / in Li(G). We show that, unlike the ideals of representable and Kalton operators, there is no subideal B of M(G) such that T is completely continuous if and only if su> G B for almost u> in Q. We actually exhibit a noncompletely continuous operator T such that su G I2+e(G) for each e > 0. I. Random measures associated to operators on h\. Let K be a separable compact Hausdorff space, A a Radon probability on K and (U, f,P) a probability space. Denote by M(K) the space of all Radon measures on K. We will call random measure on K every measurable map w —+ p^ from (U, 7, P) into M(K) when M(K) is equipped with the cr-field generated by the o(M(K), C(K)) topology. The starting point of this paper is the following disintegration theorem estab- lished by Kalton (3) and Fakhoury (2). (a) If T is a bounded linear operator from Lx(K, X) into Li(fi, 7, P), then there exists an essentially unique random measure (pj) verifying the following properties: (i) Each / in Lx(K, X) belongs to Lx(K, \p^\) for P almost all uj. (ii) Tf(w) = iK f(t) dpu(t) for P almost all w. (iii) / |/ij dP(u) Li(f2)

In the present paper the antisymmetry of the image of a function algebra under its contractive representation is characterized. A complete solution of this problem is obtained for subnormal contractive representations. Some applications, in particular, to the von Neumann functional calculus, are given.

Let W = U n I W, be an F0 subset of the unit circle of measure O and let {qn), n > 1, be a decreasing sequence with q, 1, there is such an H of norm 1 for which lim,rH(re' w) = I /n. The proof of the theorem depends on the existence of a special collection of closed sets {SA), A > 1, real, for which the function h, defined by h(x) = an + [(inf{AIx E S)) n](a+a), a"= In qn, is such that the function H(w) = exp(1/2r) f [(w + eau)/(eiu w)]h(u) du has the required properties. Some of the techniques used are similar to those developed in an earlier paper [1]. NOTATION. Let I be a fixed interval on the real line. For each Lebesgue measurable subset E of I, JEl denotes its Lebesgue measure. E' denotes the complement of E in I. The letters k, m, n and s when they occur as symbols denote natural numbers. C denotes the unit circle and D the unit disc. Purpose. The aim of this work is to prove the following theorem. THEOREM. Let W = U n= I W, be an F,J subset of the unit circle of measure 0 and let {q }, n > 1 be a strictly decreasing sequence with q1 1, there is such an H of norm 1 for which limr 1 H(rei'.) = l/n. Preliminary lemmas. Three lemmas precede the proof of the theorem. In addition, frequent reference is made to a lemma of Lusin-Menchoff. For convenience this lemma is stated first. LUSIN-MENCHOFF LEMMA. Let M1 be an arbitrary nonempty measurable subset of (0, 1). Let M2 be a closed subset of M1 consisting only of points of Received by the editors February 2, 1977. AMS (MOS) subject classifications (1970). Primary 26A24, 26A30, 30A88.

If a quadratic differential system has four singular points, these are elementary and the sum of their indices is 0 iff the quadrilateral with vertices at the singular points is convex; otherwise the sum of indices is 2 or -2. These facts and the relative positions of the two kinds of singular points are readily proved by consideration of the pencil of isoclines of the system. The theorem is originally due to Berlinskii. We refer to [1]. In that paper Coppel presents a theorem of Berlinskii with a proof by Kukles and Casanova. We give a simpler proof of purely geometrical character. THEOREM. Suppose that a real quadratic differential system has four critical points in the affine plane. If the quadrilateral with vertices at these points is convex then two opposite critical points are saddles and the other two are antisaddles (nodes, foci or centers). But if the quadrilateral is not convex, then either the three exterior points are saddles and the interior vertex an antisaddle or the exterior vertices are antisaddles and the interior vertex a saddle. Preliminaries. A real quadratic differential system is one of the form x = P(x,y), y = Q(x,y), where P(x, y) and Q(x, y) are real second degree polynomials. We assume that the curves P = 0 and Q = 0 meet at four distinct points of the affine plane, and then, the four points are simple critical points also known as elementary singular points of the system. That is because the condition of four distinct points ensures that the tangents to P = 0 and Q = 0 at each of them do not coincide, so that the determinant Px(xi,yi) x Qy(xi,yi) Py(xi,yi) x Qx(xi,yi) is different from zero, i = 1, 2, 3, 4. But this is the defining condition of an elementary singular point of the system and it is known that such point is either a saddle or an antisaddle: focus, center, node (see [2, Chapter 6, ?3]). The index of a saddle (angular variation of the field along a simple closed curve surrounding the singular point and no other) is -1, and antisaddles have index + 1. In the course of the proof we will need the fact that the sum of indices of a set of singular points is equal to the angular variation of the field along any simple closed curve surrounding those points and no others, also called the index of the curve with respect to the field (see [2, Appendix, ? 1]). These four critical points of the system are also base points of a pencil of conics consisting of P = 0 and the curves XP Q = 0, with X a real Received by the editors July 14, 1978 and in revised form, February 28, 1979. AMS (MOS) subject classifications (1970). Primary 34CO5 ? 1980 American Mathematical Society 0002-9939/80/0000-01 15/$01.75

The main aim of the paper is to investigate the relationship between certain generalised Lipschitz classes of functions and to discuss the absolute Nörlund summability of Fourier series of functions of the class L q {L^q} where 2 > q > ∞ 2 > q > \infty .

It is shown that there exist real analytic real hypersurfaces in CG which cannot be locally holomorphically imbedded in any finite dimensional sphere S2N-1 C C2N. In [1], Chern and Moser developed a theory of local invariants of real hypersurfaces in complex manifolds. In doing so, they stressed the analogies between pseudoconformal geometry and Riemannian geometry, defining analogues of the Riemannian curvature tensor, the Levi-Civita connection, geodesics, etc. These analogies are in general quite weak (e.g., "geodesics" may spiral, see [2]). The purpose of this note is to show that there is another case where the analogy breaks down. It is a classical result that any analytic metric can be locally induced by the flat metric, i.e., given any analytic metric there exists locally an isometric imbedding into some Euclidean space. The corresponding theorem in pseudoconformal geometry would state that every analytic real hypersurface could be (locally) holomorphically imbedded in the unit sphere in some Cn. The two theorems of this paper give counterexamples to this. It should be noted that if a global holomorphic imbedding of the boundary of a strictly pseudoconvex domain into a unit sphere exists, it naturally extends to a proper holomorphic mapping of the whole domain into the unit ball. Lempert [4] has shown that any smoothly bounded strictly pseudoconvex domain can be properly and holomorphically imbedded in the unit ball in infinite dimensional space (i.e., 12). The results here indicate that Lempert's theorem cannot be extended to imbeddings in finite dimensional space. THEOREM. Given integers N > n > 2, there exists a smooth, strictly pseudoconvex, real analytic (in fact, polynomial) real hypersurface M in Cn that cannot be locally analytically imbedded in the unit sphere in CN. PROOF. We shall examine the problem of mapping a real hypersurface M into CN so that the image is tangent to the sphere to order d. It will suffice to show that by taking d large enough, we can find a real hypersurface M that cannot be imbedded in CN, tangent to the unit sphere to order d. Let V be the space of real-valued polynomials F(z', z', u) of degree less than or equal to d that vanish at (0,0,0), W the space of complex polynomials f (z', w), ct = 1, ... , N, of degree less than or equal to d such that Za Ifa(0, 0)12 1 = 0. Received by the editors March 25, 1987 and, in revised form, July 6, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 32F25; Secondary 53B25, 53B15. ?1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page

Introduction. One expects that the absolutely continuous part of the spectrum of a Hamiltonian operator H= -A+ V in L2(En) (where A is the Laplacian operator and V is the operation of multiplication by a real function which approaches 0 at oo) will be the interval [0, oo). That this is the essential spectrum has been shown under very weak assumptions on V [7], but the absolute continuity has been demonstrated only under much stronger assumptions [l], [2], [3], [8]. In this paper we prove that for smooth positive potentials V which are sufficiently repulsive outside some bounded set, the operator -A+ V is absolutely continuous. Our conditions are similar to those in the previous work of Odeh [5]. We use results of Putnam [6] on commutators of pairs of selfadjoint operators. Our method works for dimensions n= 1, 2, or 3, though we consider only two cases, n= 1 (because of its simplicity) and n =3 (because of its importance for applications). Only partial results seem possible in higher dimensions.

In this note we give a generalization of a result of Aronszajn and Smith [i] concerning sections of exceptional sets for the spaces of Bessel potentials on Rn of L2 functions and restrictions of functions in these spaces. For the sake of completeness we recall briefly the relevant definitions and theorems. We refer for details to [2]. Throughout this paper we will be concerned with functions defined on the space Rn; we write f for fe, LP for LP(Rn), Co' for Co (Rn), etc. The Bessel kernel Ga on Rn of order a> 0 is defined as the inverse Fourier transform of the function

Let Qt(x) denote the number of semi-£-free integers £ix. It is known that Q*(x)=ai*x+0(x1/k), where a* is a constant. In this paper we prove that Aito = fiito - «I* = 0(x1/k cxp{-A lof^xQoglogx)-1^}), where A is an absolute positive constant. Further, on the assump- tion of the Riemann hypothesis, we prove that A*(x) = 0(jc2""+1> exp{/l log x(iog log x)~1}).

F. W. Wilson points out that the existence of a homeomorphism between the level surfaces 24 and 12 is far from trivial. In fact, he shows in a paper to appear in the Journal of Differential Equations that if dim 2;i>4 then the existence follows from the Poincare conjecture. Nothing can be said about the diffeomorphism class of Xi. Thus, the existence of the homeomorphism on 11 onto 12 must be assumed in the proof of the theorem if dim 2% = 3 or 4, in which dimensions the validity of the Poincare conjecture is not known.

The paper [1I, dealing with applications of boundary properties of the Riemann mapping function, makes essential use of a rectifiability hypothesis on the boundary. Michael R. Cullen has kindly called the author's attention to the fact that the form in which this hypothesis is stated differs from that actually used in the derivations. However, a simple rephrasing of the underlying conventions on the region Q suffices to set matters straight, and we present the necessary changes here, along with some related comments. A generalization of the classical Osgood-Taylor-Caratheodory theorem serves as the starting point in [1]. Let Q be a bounded simply connected plane region for which (iQ can be parametrized as a closed curve. Then any Riemann mapping function X for Q, i.e. any function mapping the open unit disc co conformally onto Q, can be extended to a continuous mapping of co onto Q. In stating the rectifiability hypothesis in [1] a parallel wording was used, namely that aQ be parametrizable as a rectifiable closed curve. The condition actually employed in the derivations, however, is that of rectifiability of disc-induced parametrizations of the prime-end boundary. While the two conditions are probably equivalent, a proof does not appear to be obvious. On the other hand, the whole question can be regarded as peripheral to the analytical results of [1], since these would normally be applied (as in the case of the disc slit along a radius) by tracing out the prime-end boundary in the natural way. As discussed in [1], the generalized Osgood-Taylor-Caratheodory theorem ensures that the prime-end boundary can be parametrized as a Jordan curve, the so-called Jordan-Caratheodory boundary curve A. Since any two Riemann mapping functions X are connected by a linear fractional transformation, it is clear that different choices of X yield equivalent parametrizations of A. Thus, the notions of JordanCaratheodory boundary curve and arc length along such a curve are intrinsic. In the light of these remarks we revise Theorem 1 of [1] as

The purpose of this paper is to translate the condition defining Lebesgue covering dimension of a topological space X into a condition on C(X), the ring of continuous real-valued functions on X. We take the definition of topological dimension given in [2, p. 243]. The definition in [2] is given for completely regular Hausdorff spaces, but applies equally well to arbitrary spaces. Characterizations of dim X in terms of C(X) have been given by Katetov and by the author [1]. For an exposition of Katetov's work the reader is referred to [2, Chapter 16]. Let R be a commutative ring with identity. By a basis in R we mean a finite set of elements which generate R. The order of a basis is the largest integer n for which there exist n+ 1 members of the basis with nonzero product. There is a close relation between bases in C(X) and basic covers of X [2, p. 243]. For each basis ffi}i,j of C(X) we associate the basic cover {Ui}I of X, where Ui is defined by Uj={x:-f(x)O}. Conversely, for each basic cover {Uil}fc of X, we may associate a basis {fj}jEl of C(X) where fi is chosen to satisfy Uj={x:-f(x)O}. Since U4 ...*frUi>= 0 if and only if fi * *fin=O, it follows that the order of the basic cover {Ui}lie is the same as the order of the basis {fi}fl6. If now {aj}jcI and {bj}jEJ are bases in the ring R, we say that {bj}j3j is a refinement of failic, if for each ]eJ there is an icI such that b, is a multiple of ai. The dimension of R, denoted by d(R), is here defined to be the least cardinal m such that every basis of R has a refinement of order at most m. THEOREM. If X is an arbitrary topological space, then dim X=d(C(X)). PROOF. Suppose d(C(X))_n. Let {Uj}jj1 be a basic cover of X and let {f?}jej be an associated basis in C(X). By hypothesis, this basis has a refinement {gj}jcj of order at most n. The basic cover associated with {gj}j,Ej is then a refinement of {Ui})i1 of order at most n. Thus dim X_n. Received by the editors July 20, 1971. AMS-1969 subject classifications. Primary 5470.

Presented to the Society, September 7,1948; received by the editors December 28, 1948. 1 This paper was written at the Institute for Numerical Analysis of the National Bureau of Standards, with the financial support of the Office of Naval Research of the U. S. Navy Department. * G. Szego, On an inequality of P. Turdn concerning Legendre polynomials, Bull. Amer. Math. Soc. vol. 54 (1948) pp. 401-405. 256

Several authors have recently shown that Brownian motion with continuous paths on [ 0 , 1 ] [0,1] can be expanded into a uniformly convergent (a.s.) orthogonal series in terms of a given complete orthonormal system (CONS) in its reproducing kernel Hilbert space (RKHS). In an earlier paper we generalized this result to Gaussian processes with continuous paths. Here we obtain such expansions for a Gaussian random variable taking values in an arbitrary separable Banach space. A related problem is also considered in which starting from a separable Hilbert space H H with a measurable norm | | ⋅ | | 1 || \cdot |{|_1} defined on it, it is shown that the corresponding abstract Wiener process has a | | ⋅ | | 1 ||\cdot |{|_1} -convergent orthogonal expansion in terms of a CONS chosen from H H .