A fundamental question in operator theory is: how rich is the collection of oprrators on a given Banach space? For classical spaces, especially Hilbert space, a well developed theory of operators exists. However a Banach space may have far fewer operators than one might expect. Shelah [S] has constructed a nonseparable Ba~lach space X , for which the space of operators with separable range has ...

Let X be a (real or complex) Banach space, its dual Banach space will be denoted by X *. We use standard notation and terminology of Banach space theory, see J.Lindenstrauss and L.Tzafriri [LT]. By a subspace we mean a linear, but not necessarily closed, subspace. We also assume some knowledge of general topology and ordinal numbers, see P.S.Aleksandrov[A]. Definition 1.1. Let A be a subset of ...

March 7, 2000 Abstract. A Banach space X is said to have the separable lifting property if for every subspace Y of X containing X and such that Y/X is separable there exists a bounded linear lifting from Y/X to Y . We show that if a sequence of Banach spaces E1, E2, . . . has the joint uniform approximation property and En is c-complemented in E∗∗ n for every n (with c fixed), then P n En 0 has...

It is shown that every separable Banach space admits an equivalent norm that is uniformly Gâteaux smooth and yet lacks asymptotic normal structure. A Banach space is said to have the fixed point property (FPP) if for every nonempty bounded closed convex C ⊂ X and every nonexpansive self-mapping T : C → C there is a fixed point of T in C. A Banach space is said to have the weak fixed point prope...

A necessary and sufficient condition for existence of a Banach space with a finite dimensional decomposition but without the π-property in terms of norms of compositions of projections is found. 2000 Mathematics Subject Classification. Primary 46B15; Secondary 46B07, 46B28. The problem of existence of Banach spaces with the π-property but without a finite dimensional decomposition is one of the...

We establish sufficient conditions on the shape of a set A included in the space Lns (X, Y ) of the n-linear symmetric mappings between Banach spaces X and Y , to ensure the existence of a C-smooth mapping f : X −→ Y , with bounded support, and such that f (X) = A, provided that X admits a Csmooth bump with bounded n-th derivative and dens X = densL(X, Y ). For instance, when X is infinite-dime...

Let H be a separable Banach space. We considered the sequence of stochastic integrals {Xn− · Yn} where {Yn} is a sequence of infinite dimesnional H semimartingales and Xn are H valued cadlag processes. Assuming that {(Xn, Yn)} satisfies large deviation principle, a uniform exponential tightness condition is described under which large deviation principle holds for {(Xn, Yn, Xn− · Yn)}. When H i...

Abstract. We prove that for every function f : X → Y , where X is a separable Banach space and Y is a Banach space with RNP, there exists a set A ∈ Ã such that f is Gâteaux differentiable at all x ∈ S(f) \ A, where S(f) is the set of points where f is pointwise-Lipschitz. This improves a result of Bongiorno. As a corollary, we obtain that every K-monotone function on a separable Banach space is...

This paper contains the following results for ecohomies with infinite dimensional commodity spaces. (i) We establish a core-Walras equivalence theorem for economies with an atomless measure space of agents and with an ordered separable Banach commodity space whose positive cone has a non-empty norm interior. This result includes as a special case the Aumann (1964) and Schmeidler-Hildenbrand [Hi...

Recently the author [Proc. Amer. Math. Soc. 103(1988), 11291135] proved random versions of an interesting theorem of Ky Fan [Theorem 2, Math. Z. 112 (1969), 234-240] for continuous condensing random maps and nonexpansive random maps defined on a closed convex bounded subset in a separable Hilbert space. In this paper, we prove that it is still true for (more general) continuous 1-set-contractiv...