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Ditor and Eifler consider the open mapping theorem for the probability spaces. Here we attempt to generalize the theorem to the spaces of Young measures.

In this paper we present some generalization (at the same time a new and a short proof in the Banach algebra context) of the Weak Spectral Mapping Theorem (WSMT) for non-quasianalytic representations of one-parameter groups.

We prove an interpolation theorem for Hilbert spaces of analytic functions that have the Nevanlinna-Pick property. This result applies to Dirichlet and Dirichlet-type spaces, and in particular a short proof of the theorem by Marshall-Sundberg on interpolating sequences is obtained.

Our main result is a characterization of open Seifert fibered 3 3 -manifolds in terms of the fundamental group and large-scale geometric properties of a triangulation. As an application, we extend the Seifert Fiber Space Theorem and the Torus Theorem to a class of 3 3 -orbifolds.

The pseudospectrum has become an important quantity for analyzing stability of nonnormal systems. In this paper, we prove a mapping theorem for pseudospectra, extending an earlier result of Trefethen. Our result consists of two relations that are sharp and contains the spectral mapping theorem as a special case. Necessary and sufficient conditions for these relations to collapse to an equality are demonstrated. The theory is valid for bounded linear operators on Banach spaces. For normal matrices, a special version of the pseudospectral mapping theorem is also shown to be sharp. Some numerical examples illustrate the theory.

A generalization of Phelps' lemma to locally convex spaces is proven, applying its well-known Banach space version. We show the equivalence of this theorem, Ekeland's principle and Danes' drop theorem in locally convex spaces to their Banach space counterparts and to a Pareto efficiency theorem due to Isac. This solves a problem, concerning the drop theorem, proposed by G. Isac in 1997. We show that a different formulation of Ekeland's principle in locally convex spaces, using a family of topology generating seminorms as perturbation functions rather than a single (in general discontinuous) Minkowski functional, turns out to be equivalent to the original version.

The Bartle-Graves theorem extends the Banach open mapping principle to a family of linear and bounded mappings, thus showing that surjectivity of each member of the family is equivalent to the openness of the whole family. In this paper we place this theorem in the perspective of recent concepts and results, and present a general Bartle-Graves theorem for set-valued mappings. As application, we obtain versions of this theorem for mappings defined by systems of inequalities, and for monotone variational inequalities.

We consider real spaces only. Definition. An operator T : X → Y between Banach spaces X and Y is called a Hahn-Banach operator if for every isometric embedding of the space X into a Banach space Z there exists a norm-preserving extension T of T to Z. A geometric property of Hahn-Banach operators of finite rank acting between finite-dimensional normed spaces is found. This property is used to characterize pairs of finite-dimensional normed spaces (X, Y ) such that there exists a Hahn-Banach operator T : X → Y of rank k. The latter result is a generalization of a recent result due to B. L. Chalmers and B. Shekhtman. Everywhere in this paper we consider only real linear spaces. Our starting point is the classical Hahn-Banach theorem ([H], [B1]). The form of the Hahn-Banach theorem we are interested in can be stated in the following way. Hahn-Banach Theorem. Let X and Y be Banach spaces, T : X → Y a bounded linear operator of rank 1 and Z a Banach space containing X as a subspace. Then there exists a bounded linear operator T : Z → Y satisfying (a) ||T || = ||T ||; (b) T x = Tx for every x ∈ X. Definition 1. An operator T : Z → Y satisfying (a) and (b) for a bounded linear operator T : X → Y is called a norm-preserving extension of T to Z. The Hahn-Banach theorem is one of the basic principles of linear analysis. It is quite natural that there exists a vast literature on generalizations of the HahnBanach theorem for operators of higher rank. See papers by G. P. Akilov [A], J. M. Borwein [Bor], B. L. Chalmers and B. Shekhtman [CS], G. Elliott and I. Halperin [EH], D. B. Goodner [Go], A. D. Ioffe [I], S. Kakutani [Kak], J. L. Kelley [Kel], J. Lindenstrauss [L1], [L2], L. Nachbin [N1] and M. I. Ostrovskii [O], representing different directions of such generalizations, and references therein. There exist two interesting surveys devoted to the Hahn-Banach theorem and its generalizations; see G. Buskes [Bus] and L. Nachbin [N2]. We shall use the following natural definition. Definition 2. An operator T : X → Y between Banach spaces X and Y is called a Hahn-Banach operator if for every isometric embedding of the space X into a Banach space Z there exists a norm-preserving extension T of T to Z. Received by the editors February 9, 2000. 2000 Mathematics Subject Classification. Primary 46B20, 47A20.

"Reduction" of linear operators is effected by commuting projections; the spectrum of the operator is then the union of the spectra of its range and null space restrictions. Disjointness of these partial spectra implies that the projection "double commutes" with the operator, which in turn can be recognised as a curious kind of "exactness". Variants of this exactness correspond to various kinds of disjointness between the partial spectra. INTRODUCTION "Reduction" of an operator T on a linear space X means writing X = Xo ff X1 as the direct sum of a pair of complementary invariant subspaces TXj C Xj for T: the philosophy is that each of the induced operators Tj is in some sense simpler than the original operator T, whose behaviour on X can be reconstructed from the behaviour of To and T1. When X is a Banach space and T is continuous it is natural to ask that the subspaces Xj be closed also; then the open mapping theorem says that there is a bounded projection (0.1) Q = Q2 C BL(X, X) with QT = TQ, commuting with T, whose range and null space furnish the subspaces X1 = Q(X) and XO = Q-1(0). In this situation the behaviour of T is rather easily recovered from T1 and To; for example the spectrum a(T) = u(Ti) U a(To). Of course it is not at all clear that such reduction can be achieved in a non-trivial way: the operator T may have no invariant subspaces other than {0} and X, such invariant subspaces may not have closed complements, and if they are complemented, they may not have invariant complements. One situation in which such a projection certainly can be found is when the spectrum of T can be written as a disjoint union of closed subsets 5(T) = K, U Ko with K n Ko =0. Received by the editors December 15, 1997 and, in revised form, March 10, 1998, October 6, 1998, and March 31, 1999. 1991 Mathematics Subject Classification. Primary 47A13; Secondary 15A21, 15A18.

We consider a linear sytem in a Banach lattice and provide a simple theorem of the alternative (or Farkas lemma) without the usual closure condition.

The least cardinal for which the Baire category theorem fails is equal to the least cardinal for which a Ramseyan theorem fails. The Baire category theorem states that the real line is not the union of countably many meager (also known as first category) sets. Let cov(M) denote the least cardinal number such that there are that many first category subsets of the real line whose union is the entire real line. Then cov(M) is the least cardinal number for which the Baire category theorem fails. This cardinal number, defined in terms of topological notions, appears in many different guises in combinatorial set theory. A long list of diverse guises of this cardinal number is already general knowledge for set theorists; Galvin gave a game-theoretic version (part of which is published in [4], and part of which is unpublished—however, see [8]), A. W. Miller gave a characterization in terms of sequences of positive integers [6], which was later given an elegant improvement by Bartoszynski [1]. It is also known to be the least cardinal number such that there is a set of real numbers of that cardinality which does not have Rothberger’s property C′′. A set X of real numbers has property C′′ if, for every sequence (Un : n = 1, 2, 3, . . . ) of open covers of X , there is a sequence (Un : n = 1, 2, 3, . . . ) such that, for each n, Un ∈ Un and {Un : n = 1, 2, 3, . . .} is a cover for X . It seems that for the purposes of applications of set theory to other areas of mathematics, it would be useful to have as many non-trivial characterizations of this cardinal number as possible. In this paper we give a few more equivalent forms of this cardinal number. To explain some of our results, we need some terminology which is well-known in other contexts. Let κ be an infinite cardinal number which will be fixed for the duration of the paper. A collection of subsets of κ is said to be a cover of κ if its union is equal to κ. We shall be interested in countable covers of κ. A cover of κ is said to be an ω-cover if it is countably infinite, κ itself is not a member of the cover, and if there is for every finite subset of κ an element of this cover which contains it. We shall let the symbol Ω denote the collection of ω-covers of κ. Borrowing from Ramsey theory (see Section 8 of [3]), we shall use the symbol

We lift upper and lower estimates from linear functionals to nhomogeneous polynomials and using this result show that loo is finitely represented in the space of n-homogeneous polynomials, n > 2, on any infinitedimensional Banach space. Refinements are also given. The classical Dvoretzky spherical sections theorem [5, 13] states that 12 is finitely represented in any infinite-dimensional Banach space. Using this, the Riesz Representation theorem (for finite-dimensional 4p spaces) and the HahnBanach theorem, we show that lo is finitely represented in .(nE), for any infinite-dimensional Banach space and any n > 2. This shows that 9(nE) does not have any non-trivial superproperties and explains why spaces such as Tsirelson's space play such a positive role in the recent theory of polynomials on Banach spaces ([1, 2, 6, 7, 8, 9, 10]). We refer to [3, 11, 12] for properties of Banach spaces and to [4] for properties of polynomials. Theorem 1. Suppose E is a Banach space, 1 q, where + =1, and any sequence of scalars (a )k=1 we have (2) An sup kc'W? Z

Let X X be a uniformly convex Banach space with a Frechet differentiable norm, C C a bounded closed convex subset of X X , and T : C → C T:C \to C an asymptotically nonexpansive mapping. It is shown that for each x x in C C , the sequence { T n x } \{ {T^n}x\} is weakly almost-convergent to a fixed point y y of T T , i.e., ( 1 / n ) ∑ i = 0 n − 1 T k + i x → y (1/n)\sum \nolimits _{i = 0}^{n - 1} {{T^{k + i}}x \to y} weakly as n n tends to infinity uniformly in k = 0 , 1 , 2 , … k = 0,1,2, \ldots

Let A A be a symmetric Banach ∗ ^* -algebra, let B B be a Banach algebra, and assume that A ⊆ B A\subseteq B . A result is proved giving conditions which imply that every element of A A has the same spectrum in both A A and B B .

We characterize functions of the first Baire class with values in Banach spaces and give a short self-contained proof of a result more general than the following : if T is a complete metric space, X is a Banach space, and Φ:T→γ(X) (the power set of X) is a mapping that is usc in the weak topology then Φ has a selector of the first Baire class. This extends some results of Hansell, Jayne, Rogers, and Talagrand

A theorem is proved which generalizes both the Vietoris-Begle theorem and the cell-like theorem for spaces of finite defomation dimension. The proof is geometric and uses a double mapping cylinder trick.

An example is given of an unbounded operator in a Hilbert space which generates a strongly continuous semigroup and for which the spectral mapping theorem for the exponential function does not hold. The spectra of both the generator and the semigroup are determined explicitly.

Cohen [4] gave an example of a surjective bilinear mapping between Banach spaces which was not open, and Horowitz [8] gave a much simpler example. We build on Horowitz’ example to produce a similar result for bilinear mappings such that every element of the target space is a linear combination of n n elements of the range. An immediate application is that Bercovici’s construction [1] of an operator algebra with property ( A 1 ) ({\mathbb {A}_1}) but not ( A 1 ( r ) ) ({\mathbb {A}_1}(r)) can be extended to achieve property ( A 1 / n ) ({\mathbb {A}_{1/n}}) without ( A 1 / n ( r ) ) ({\mathbb {A}_{1/n}}(r)) .

Let F be a quasi-complete locally convex space, ( Ω , Σ , μ ) (\Omega ,\Sigma ,\mu ) a complete probability space, and L 1 ( μ ; F ) {L^1}(\mu ;F) the space of all strongly integrable functions f : Ω → F f:\Omega \to F with the Egoroff property. If F is a Banach space, then the Radon-Nikodým theorem was proved by Rieffel. This result extends to Fréchet spaces. If F is dual nuclear, then the Lebesgue-Nikodým theorem for the strong integral has been established. However, for nonmetrizable, or nondual nuclear spaces, the Radon-Nikodým theorem is not available in general. It is shown in this article that the Radon-Nikodým theorem for the strong integral can be established for quasi-complete locally convex spaces F having the following property: (CM) For every bounded subset B ⊂ l N 1 { F } B \subset l_N^1\{ F\} , the space of absolutely summable sequences, there exists an absolutely convex compact metrizable subset M ⊂ F M \subset F such that Σ i = 1 ∞ p M ( x i ) > 1 , ∀ ( x i ) ∈ B \Sigma _{i = 1}^\infty {p_M}({x_i}) > 1,\forall ({x_i}) \in B . In fact, these spaces have the Radon-Nikodým property, and they include the Montel (DF)-spaces, the strong duals of metrizable Montel spaces, the strong duals of metrizable Schwartz spaces, and the precompact duals of separable metrizable spaces. When F is dual nuclear, the Radon-Nikodým theorem reduces to the Lebesgue-Nikodým theorem. An application to probability theory is considered.

We discuss the problem of preservation of Baire category under continuous and feebly open preimages. We obtain a solution by imposing a completeness condition on the fibres f − 1 ( y ) {f^{ - 1}}(y) of the function f f under consideration. Based on a theorem on the invariance of residuality under continuous and nearly feebly open images, we also derive a result concerning the preservation of category under continuous and nearly feebly open preimages. We end up with an open mapping theorem for functions f f of this kind defined on a Čech complete space.

Let κ \kappa be the least cardinal such that the real line can be covered by κ \kappa many nowhere dense sets. We show that κ \kappa can be characterized as the least cardinal such that "infinitely equal" reals fail to exist for families of cardinality κ \kappa .

A topological space satisfying the open mapping theorem is called a B r {B_r} -space. We investigate the question whether completely regular B r {B_r} -spaces must be Baire spaces. The answer we obtain is twofold and surprising. On the one hand there exist first category completely regular B r {B_r} -spaces. Examples are provided in the class of Lindelöf P P -spaces. On the other hand, we obtain a partial positive answer to our question. We prove that every suborderable metrizable B r {B_r} -space is in fact a Baire space. We conjecture that this is true for metrizable B r {B_r} -spaces in general. Our paper is completed by some applications. For instance, we establish the existence of a metrizable B r {B_r} -space E E whose square E × E E \times E is no longer a B r {B_r} -space.

The Wilansky-Klee conjecture is equivalent to the (unproved) conjecture that every dense, 1-codimensional subspace of an arbitrary Banach space is a Baire space (second category in itself). The following two characterizations may be useful in dealing with this conjecture: (i) A topological vector space is a Baire space if and only if every absorbing, balanced, closed set is a neighborhood of some point. (ii) A topological vector space is a Baire space if and only if it cannot be covered by countably many nowhere dense sets, each of which is a union of lines (1-dimensional subspaces). Characterization (i) has a more succinct form, using the definition of Wilansky's text [8, p. 224]: a topological vector space is a Baire space if and only if it has the t property. Introduction. The Wilansky-Klee conjecture (see [3], [7]) is equivalent to the conjecture that every dense, 1-codimensional subspace of a Banach space is a Baire space. In [4], [5], [6], [7] it is shown that every countable-codimensional subspace of a locally convex space which is "nearly" a Baire space is, itself, "nearly" a Baire space. (Theorem 1 of this paper indicates how "nearly Baire" a barrelled space is: Wilansky's class of W-barrelled spaces [9,p. 44] is precisely the class of linear Baire spaces.) The theorem [7] that every countable-codimensional subspace of an unordered Baire-like space is unordered Baire-like is the closest to an affirmation of the conjecture. (A locally convex space is unordered Baire-like if it cannot be covered by countably many rare [nowhere dense], balanced, convex sets.) The two characterizations in this note, of some independent interest, seem also likely aids for tackling the Wilansky-Klee conjecture. (E.g., see Remarks (a).) Presented to the Society, January 15, 1974; received by the editors January 29, 1973. AMS (MOS) subject classifications (1970). Primary 46A15; Secondary 46A07.

Let E 1 , E 2 {E_1},{E_2} be two Banach spaces, and let f : E 1 → E 2 f:{E_1} \to {E_2} be a mapping, that is “approximately linear". S. M. Ulam posed the problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist". The purpose of this paper is to give an answer to Ulam’s problem.

If the spaces C ( T , R ) C(T,R) and C ( S , R ) C(S,R) of continuous functions on S S and T T are linearly isometric, then T T and S S are homeomorphic. By the classical Stone-Banach theorem the only linear isometries of C ( T , R ) C(T,R) onto C ( S , R ) C(S,R) are of the form x → a ( x ∘ h ) x \to a(x \circ h) , where h h is a homeomorphism of S S onto T T and a ∈ C ( S , F ) a \in C(S,F) is of magnitude 1 for all s s in S S . What happens if R R is replaced by a field with a valuation? In brief, the result fails. We discuss "how" by way of developing a necessary and sufficient condition for the theorem to hold, along with some examples to illustrate the point.

Applying a theorem of Bagemihl and Seidel (1953), we prove that if S 2 {S_2} is a set of second category in ( α , β ) (\alpha ,\beta ) , where 0 ⩽ α > β ⩽ 2 π 0 \leqslant \alpha > \beta \leqslant 2\pi , and if f ( z ) f(z) is a function meromorphic in the sector Δ ( α , β ) = { z : 0 > | z | > ∞ , α > arg ⁡ z > β } \Delta (\alpha ,\beta ) = \{ z:0 > \left | z \right | > \infty ,\alpha > \arg z > \beta \} for which lim _ r → ∞ | f ( r e i θ ) | > 0 {\underline {{\operatorname {lim}}} _{r \to \infty }}\left | {f(r{e^{i\theta }})} \right | > 0 , for all θ ∈ S 2 \theta \in {S_2} , then there exists a sector Δ ( α ′ , β ′ ) ⊆ Δ ( α , β ) \Delta (\alpha ’,\beta ’) \subseteq \Delta (\alpha ,\beta ) such that ( α ′ , β ′ ) ⊆ S ¯ 2 , S 2 (\alpha ’,\beta ’) \subseteq {\bar S_2},{S_2} is second category in ( α ′ , β ′ ) (\alpha ’,\beta ’) , and f ( z ) f(z) has no zero in Δ ( α ′ , β ′ ) \Delta (\alpha ’,\beta ’) . Based on this property, we prove several uniqueness theorems.

We prove graph theorems and a variant of the Banach-Steinhaus theorem in a purely topological context. We obtain characterizations of Baire spaces in the class of metrizable spaces by means of graph theorems. Introduction. The following abstract form of a graph theorem covers the preponderant part of work in this field: Let f be a nearly continuous mapping from a space E to a space F which satisfies a certain condition (frequently expressed in terms of the graph of f ). Then, under appropriate conditions on E and F, the mapping f is continuous. The most convenient applications of this scheme deal with the case where the graph of the mapping f is closed. The reader might consult [Ko, p. 33ff], where the classical theory is treated, [Hu] for the situation in topological groups, [Ke], [LR] for the case of uniform spaces, [Wi], [BP], [We] for the general topological case. In this paper, the main interest is directed toward the case where the graph G(f ) of f is a G6-set (more generally a set of interior condensation). In ?1 we prove that if E is a Baire space and if F is complete (in a certain sense) then the graph theorem above holds for mappings f whose graph is G6. In ?2 we prove graph theorems for mappings with measurability conditions. In this case the completeness property for F is no longer needed. As a consequence, in ?2 we obtain a purely topological variant of the Banach-Steinhaus theorem. ?3 is devoted to the study of a converse problem arising from our graph theorems in ??1 and 2. In view of the fact that these theorems hold for source spaces E which are Baire spaces it is natural to pose the following question: Let E be a space such that a certain graph theorem holds for E and all suitable f and F. Must E then be a Baire space? We prove that the answer is in the positive if E is assumed to be metrizable, thus obtaining three characterizations of Baire spaces in the framework of metrizable topological spaces. Finally we prove that in the absence of metrizability the Baire condition is no longer necessary. In the final ?4 we give a brief examination of a graph theorem of Wilhelm's [Wi], in which almost continuous mappings are used instead of the nearly continuous ones used here. Received by the editors November 7, 1983 and, in revised form, February 4, 1985. 1980 Mathematics Subject Classification. Primary 54E52; Secondary 54C10.

Only the usual axioms of set theory are needed to prove the existence of a Baire space whose square is not a Baire space. Assuming the continuum hypothesis (CH), Oxtoby [9] constructed a Baire space whose square is not Baire. We will show in this paper that the assumption of CH is unnecessary. Such results are greatly enhanced by Krom [5], who showed that if there is such an example, then there is also a metric example. Remarks of the referee were instrumental in making the main result of this paper an absolute one rather than one of relative consistency. In particular, the author was not aware of the forcing technique of ?2. Comments on this paper by Franklin Tall were also of great help to the author. 1. Baire spaces and forcing. Suppose P = is a partially ordered structure. P may be regarded as a topological space where the initial segments of VP generate a basis. If P and 2 are partially ordered sets, then the Cartesian product P x Q may be partially ordered pointwise to obtain a partially ordered set 'Y x B. It is easily seen that P x 2, considered as a topological space, is homeomorphic to the product of topological spaces VP and B. A topological space is said to be Baire if any countable intersection of its dense open sets is dense. If the space is derived from a partially ordered set as above, then we note that any such countable intersection is necessarily open. Two elements of a partially ordered set will be called compatible if they have a common predecessor. A partially ordered structure P = will be called fine if for every p, q E P, either (1) q _ p or (2) there is an r _ q which is incompatible with p. Suppose Y1 is a countable standard transitive model of Zermelo-Fraenkel set theory (ZFC) and P, 2 E Y1 are partially ordered sets. We collect below some well-known facts. LEMMA 1.0. If P is fine, then the following statements are equivalent. (a) P is Baire in T (b) Whenever G is 'P-generic over Y1 and f E 61[G] is an ordinal valued function with domain w, then f E (=-6. Received by the editors June 13, 1975. AMS (MOS) subject classifications (1970). Primary 02K25, 04A30; Secondary 08A10, 54B 10, 54G20.

Some extensions of the classical Borsuk-Ulam Theorem are proved by computing a bound on the homology of certain spaces similar to configuration spaces. The Bourgin-Yang Theorem and a generalization due to Munkholm are special cases of these results.

First, in the second section, we describe a class of Baire spaces for which Blumberg's theorem does not hold. Then, in the third section, we discuss Blumberg's theorem for P-spaces. 1. In [2], J. C. Bradford and C. Goffman proved that a metrizable space X is a Baire space if and only if the following statement, called Blumberg's theorem, holds. 1. 1 If / is a real valued function defined on X, then there is a dense subset D of X such that fID is continuous. It follows from their proof that every topological space for which 1.1 holds is a Baire space. In [15], the author gave several examples of comHo pletely regular, Hausdorff, Baire spaces for which, if 2 = 1, 1.1 does not hold (see also [13], [14]). In ?2 we establish, using a lemma from [14], a result which shows that there are a number of Baire spaces for which 1.1 does not hold. 2. For any topological space X, we denote the weight of X, the pseudoweight of X, the density character of X, and the ring of all bounded real valued, continuous functions defined on X by wX, TrwX, AX, and C*(X), respectively (see [4, p. 619]). For any subset A of X, we denote the closure of A by cl A. We denote the set of all real numbers by R. 2.1 Theorem. Suppose X is a Baire space of cardinality 2 ? such that (a) X satisfies the countable chain condition, (b) wX = X= 2,and (c) every set of the first category in X is nowhere dense in X. Then 1.1 does not hold for X. Proof. Let 3 denote a base for the topology Jf on X of cardinality 2 0 such that 0, X e B. 'We may assume that %I is closed under countable union. Received by the editors August 30, 1973 and, in revised form, May 17, 1974. AMS (MOS) subject classifications (1970). Primary 54C30, 54F99, 54G20.

An interpolation theorem for analytic families of operators on some quasi-Banach spaces is proved. The result is applicable to spaces whose quasi-norm is defined by means of a maximal function, for example the various HP spaces on locally compact groups. Introduction. Analytic families of operators serve as an important tool in har- monic analysis. In the framework of Banach spaces they can be defined relatively easily using duality. In the context of quasi-Banach spaces this approach is not available. The importance of some quasi-Banach spaces, for example Hp spaces, justifies the development of a corresponding theory even though certain obstacles, such as the absence of a maximum principle for quasi-Banach space valued func- tions, seem to preclude one from achieving the elegance of the theory in the Banach case. (Note, however, that the apparent simplicity of the Banach space case can be misleading; see e.g. (3).) The case in which Tza takes values in a finite-dimensional subspace of the range for each a in a dense subset of ,4o H A\ has been treated in (9). The results of (9) are adequate for handling, for example, multipliers on compact groups. In the present paper we trade the finite-dimensionality condition for various other conditions on the spaces and operators and offer a theorem which can conveniently handle multipliers on noncompact groups. Unlike (9) our results here generalize the THE INTERPOLATION THEOREM. Denote the strip {z E C|0 < rez < 1} by S. The Poisson kernel for S will be denoted by dP(c,z), with c E dS and z E S. Let (Bo,Bi) be an interpolation couple of quasi-Banach spaces. Let B be a linear space which is contained in Bo + ??i. We assume that the elements of B are measurable functions on some measure space fi. We introduce a space M(B) of B-valued functions h(c) — h(c,oj) defined for almost all c E dS. We assume that M(B) is closed under pointwise multiplication by functions ip(c) which are boundary values of H°°(S) functions. The elements of M(B) must also satisfy

In a recent paper E. J. McShane [3]2 has given a theorem which is the common core of a variety of results about Baire sets, Baire functions, and convex sets in topological spaces including groups and linear spaces. In general terms his theorem states that if 7 is a family of open maps defined in one topological space X1 into another, X2, the total image 7(S) of a second category Baire set S in X1 has, under certain conditions on 7 and S, a nonvacuous interior. The point of these remarks is to show that his argument yields a theorem for a larger class than the second category Baire sets. From this there follow sligh4ly stronger and more specific versions of some of his results, including his principal theorem, as well as a proof that if S is a subset of a weak sort of topological group and S contains a second category Baire set, then the identity element lies in the interior of both S-1S and SS-1. There is also at the end an extension of Zorn's theorem on the structure of certain semigroups. In a topological space X let the closure and interior of a set E be denoted by E* and EO and the null set by A. For any set S let I(S) - U[GIG open, GnS is first category] and II(S)=X-I(S), and let III(S) be the open set II(S)0?I(X-S). By a fundamental theorem of Banach [2], SnI(S)* is first category and hence S is second category if and only if II(S)O7A. From these we note that if N is a non-null open subset of III(S), then N-S is in the first category set (X - S)G'iI(X - S), and NnQS cannot be first category since N is non-null open and disjoint with I(S). This gives us the following lemma. LEMMA 1. For any non-null open subset N of III(S), the sets N- S

containing set, for the spectrum of the operator on the interpolation space in terms of the spectra of the operator on the given two spaces. This upper bound is the best possible one that depends only on the given two spectra. Theorem 6.10 shows that the upper bound can be improved if more is known about the operator. In ?7 we give an example, 7.3, to show that the bound in Theorem 6.7 need not be attained. The example also shows that Theorem 6.10 is stronger than Theorem 6.7. ??2, 3 and 4 are mainly lemmas and definitions needed in the later sections; Theorem 2.7, which is a generalization of the three lines theorem due to Calderon, will be an important tool. A similar-type generalization of the three lines theorem is given in Lemma 5.3, which may also be of independent interest. In ?2 we give the definition of analytic interpolation, which was first introduced by A. P. Calderon at the conference on functional analysis in the memory of S. Banach, in Warsaw, 1960; we also state several of the relevant properties of analytic interpolation. In ?3 we introduce the notion of a RIS, which is essentially the way in which two commutative Banach algebras must be related so that their structure spaces interpolate in the natural manner. We also obtain some basic properties of a RIS which are needed later. In order to obtain our main theorem it seems to be necessary to compare analytic interpolation with what we call direct interpolation, which we define in ?4. The theorem of ?4 contains results which are of use in ?5. This theorem would be elementary except for the fact that the proof requires Calderon's generalized three lines theorem mentioned above.

This paper contains a Banach space generalization of Bernstein's Theorem. An absolutely monotonic (f(n) > 0, n = 1, 2, ) real valued function of a real variable is real analytic. For some background on this theorem, see S. Bernstein [1], D. V. Widder [5], or R. Boas [2]. We first present some definitions. We denote a Banach space over the reals by E; U is an open subset of E; f is a real valued function whose domain is U and Dkfg denotes the kth derivative of f evaluated at x where k=O, 1, 2, ... [4]. A function f is analytic at x in U if there exists r> 0 such that the Taylor series for f at x

Several theorems are known for extending consistent families of measures to an inverse limit or product space [1]. In this paper the notion of a consistent family of measures is generalized so that, as with general product measures [2], the spaces are not required to be of unit measure or even σ \sigma -finite. The general extension problem may be separated into two parts, from finite to countable product spaces and from countable to uncountable product spaces. The first of these is discussed in [3]. The present paper concentrates on the second. The ultimate virtual identity of sets is defined and used as a key part of the generalization and nilsets similar to those of general product measures [2] are introduced to assure the measurability of the fundamental covering family. To exemplify the extension process, it is applied to product measures to obtain a general product measure. The paper is presented in terms of outer measures and Carathéodory measurability; however, some of the implications in terms of measure algebras should be obvious.

In this paper it is shown that an operator defined on a suitable ordered Banach space of measurable functions by a positive, irreducible kernel is never quasi-nilpotent, thus giving an extension of Ando-Krieger's theorem for operators defined on ordered Banach spaces. Roughly speaking, Ando-Krieger's theorem says that positive, irreducible kernel operators on some Banach spaces of measurable functions are never quasi-nilpotent [6, V. 6.5, or 10, Theorem 136.9]. Recent extensions of Ando-Krieger's theorem in the framework of Banach lattices have been given in [3 and 5] (see also [4, 8]). It is our purpose in this short note to show that a similar result holds for kernel operators acting in more general ordered vector spaces including Sobolev spaces defined on sufficiently smooth domains of R'. For an interesting application of this kind of result see, for example, [1]. Following H. H. Schaefer [7], we always assume that the positive cone of an ordered Banach space is closed. For other definitions and terminology we refer to [7]. From now on we assume that (X, E, ,) is a finite measure space and denote by L?(X, E, ,u), or simply Lo, the set of all ,u-measurable functions on (X, Z, ,u). The cone of all positive ,u-measurable functions will be denoted by Lo+. Recall that the family of set {V,: n = 1,2,3,...}, where V,, = {f E L?: ,u(x: If(x)I > n-1) 0 for all S E E such that ,u(S) > 0, ,u(X S) > 0. We give now our extension of Ando-Krieger's theorem. The key for the proof is H. H. Schaefer's approach to the proof of this theorem [6, V.6.5 and Lemma V.6.4]. THEOREM 1. Let H be an ordered Banach space, H C Lo, with a nontrivial positive cone C = L?+ n H o {0}. Let T: H -* H be a bounded linear operator on H induced by a positive, irreducible kernel t(., .), i.e. given by Tf(y) = f t(x, y)f(x) dyu(x), f CH, Received by the editors December 1, 1986 and, in revised form, March 5, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 47B55; Secondary 47A10.

The geometric realization of the irreducible square integrable representations for semisimple Lie groups (cf. [3], [6] ) and also for nilpotent Lie groups [5] suggests that, as a general phenomenon, such representations should appear as L -kernels of invariant elliptic operators. One basic problem in this respect is to decide when such a kernel is nonzero. In the compact case the basic tool for this, used in the Borel-Weil-Bott approach, is the Hirzebruch-Riemann-Roch theorem. In the noncompact case one needs an analogue of the index theorem of Atiyah-Singer [2] for noncompact manifolds. When G possesses a discrete cocompact subgroup, the L-index theorem for covering spaces of [1] and [7] provides the required analogue. Our purpose here is to give a general index theorem for homogeneous spaces of arbitrary connected unimodular Lie groups, essentially based on the index theorem for foliations [4]. So let G be a connected unimodular Lie group, and let H be a closed subgroup of G which contains the center Z of G and such that H/Z is compact. Let x be a character of Z, and let E, F be finite-dimensional unitary representations of H whose restrictions to Z are given by xDenote by E, F the corresponding (invariant) induced bundles on the homogeneous space M = G/H, and let D be an invariant elliptic differential operator from E to F. The representation of G in the kernel of D in L(M, E) is square integrable modulo the center of G (see [4]), though not necessarily irreducible. Its formal degree deg(Ker D) (as defined in [4]) is always finite, so that the analytical index of D can be defined as

The following theorem is due to S. Piccard [2, p. 30]: "The difference of two second category Baire sets (see [1]) contains a nonempty open set". For various generalizations of this result the reader is referred to [3] and [4], where he can also find some more references. In this note we give a short proof of a generalization of Piccard's theorem. Letf: X x X -X. We definef: X -X and: X -X byfx(y)=f(x,y) and fy(x) = f(x, y) for all x, y E X. Then f is globally solvable, if f is continuous and if there exist two continuous functions 41, 4): X X X--> X such that f(x, y) = z is equivalent to x = 4(y, z) and y = ?(x, z) for all x, y, z E X. It follows that fx,P, p,A, 4)Z are homeomorphisms. If X is a topological group and f(x, y) = x *y, then 41(y, z) and +(x, z) may be taken to be z y-' and xz.