A problem of Banach asks whether every infinite-dimensional Banach space which is isomorphic to all its infinite-dimensional subspaces must be isomorphic to a separable Hilbert space. In this paper we prove a result of a Ramsey-theoretic nature which implies an interesting dichotomy for subspaces of Banach spaces. Combined with a result of Komorowski and TomczakJaegermann, this gives a positive...

We develop a theory of Malliavin calculus for Banach space valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener-Itô isometry to Banach spaces. In the white noise case we obtain two sided L-estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener-I...

A logarithmic residue is a contour integral of the (left or right) logarithmic derivative of an analytic Banach algebra valued function. Logarithmic residues are intimately related to sums of idempotents. The present paper is concerned with logarithmic residues and sums of idempotents in the Banach algebra generated by the compact operators and the identity in the case when the underlying Banac...

In this paper, we first introduce a class of nonlinear mappings called generic generalized nonspreading which contains the class of generalized nonspreading mappings in a Banach space and then prove an attractive point theorem for such mappings in a Banach space. Furthermore, we prove a mean convergence theorem of Baillon’s type and a weak convergence theorem of Mann’s type for such nonlinear m...

In [K.-I. Mitani and K.-S. Saito, J. Math. Anal. Appl. 327 (2007), 898–907] we characterized the strict convexity, uniform convexity and uniform non-squareness of Banach spaces using ψ-direct sums of two Banach spaces, where ψ is a continuous convex function with some appropriate conditions on [0, 1]. In this paper, we characterize the Bn-convexity and Jn-convexity of Banach spaces using ψ-dire...

We show that if T is an isometry (as metric spaces) between the invertible groups of unital Banach algebras, then T is extended to a surjective real-linear isometry up to translation between the two Banach algebras. Furthermore if the underling algebras are closed unital standard operator algebras, (T (eA)) −1 T is extended to a surjective real algebra isomorphism; if T is a surjective isometry...

The rate of convergence for an almost surely convergent series of Banach space valued random elements is studied in this paper. As special cases of the main result, known results are obtained for a sequence of independent random elements in a Rademacher type p Banach space, and new results are obtained for a martingale difference sequence of random elements in a martingale type p Banach space a...

We prove that a Schauder frame for any separable Banach space is shrinking if and only if it has an associated space with a shrinking basis, and that a Schauder frame for any separable Banach space is shrinking and boundedly complete if and only if it has a reflexive associated space. To obtain these results, we prove that the upper and lower estimate theorems for finite dimensional decompositi...

0. Introduction 2 The impact of logic in Banach space theory 2 The case of model theory 2 Model theory for structures of functional analysis 3 Two famous applications 4 A note on the exposition 4 1. Preliminaries: Banach Space Models 5 Banach space structures and Banach space ultrapowers 5 Positive bounded formulas 7 Approximate satisfaction 8 (1 + )-isomorphism and (1 + )-equivalence of struct...

In this note, unless we say otherwise every vector space or algebra we speak about is over C. If A is a Banach algebra and e ∈ A satisfies xe = x and ex = x for all x ∈ A, and also ‖e‖ = 1, we say that e is unity and that A is unital. If A is a unital Banach algebra and x ∈ A, the spectrum of x is the set σ(x) of those λ ∈ C for which λe−x is not invertible. It is a fact that if A is a unital B...