It will be shown and induced that the d-dimensional indices in the Banach spaces version conditions ∑ n(E‖Xn‖/|n|) < ∞ are sufficient to yield limmin1≤ j≤d(nj)→∞(1/ |nα|)∑k≤n ∏d j=1(1− (kj − 1)/nj)Xk = 0 a.s. for arrays of James-type orthogonal random elements. Particularly, it will be shown also that there are the best possible sufficient conditions for multi-indexed independent real-valued ra...

Let V be an arbitrary system of weights on an open connected subset G of CN (N ≥ 1) and let B(E) be the Banach algebra of all bounded linear operators on a Banach space E. LetHVb (G,E) andHV0 (G,E) be the weighted locally convex spaces of vector-valued analytic functions. In this survey, we present a development of the theory of multiplication operators and composition operators from classical ...

Let V be an arbitrary system of weights on an open connected subset G of CN (N ≥ 1) and let B (E) be the Banach algebra of all bounded linear operators on a Banach space E. Let HVb (G, E) and HV0 (G, E) be the weighted locally convex spaces of vector-valued analytic functions. In this paper, we characterize self-analytic mappings φ : G → G and operator-valued analytic mappings Ψ : G → B (E) whi...

We characterize compact composition operators on real Banach spaces of complex-valued bounded Lipschitz functions on metric spaces, not necessarily compact, with Lipschitz involutions and determine their spectra.

For rather general moduli of smoothness ρ, like ρ(h)=h ln(c/h), we consider the Hölder spaces Hρ(B) of functions [0, 1] → B where B is a separable Banach space. We establish an isomorphism between Hρ(B) and some sequence Banach space. With this analytical tool, we follow a very natural way to study, in terms of second differences, the existence of a version in Hρ(B) for a given random field. 20...

The Laplace transform theory of Banach space valued functions and the field of evolution equations presently offer many research problems which are of general interest in mathematical analysis. Although there is a longstanding relationship between the two fields, it was not until recently that the basic transform principles for Banach space valued functions could be formulated in a satisfactory...

We study conical square function estimates for Banach-valued functions, and introduce a vector-valued analogue of the Coifman–Meyer–Stein tent spaces. Following recent work of Auscher–MIntosh–Russ, the tent spaces in turn are used to construct a scale of vector-valued Hardy spaces associated with a given bisectorial operator A with certain off-diagonal bounds, such that A always has a bounded H...

In this paper we consider a class of distributed parameter systems (partial differential equations) determined by strongly nonlinear operator valued measures in the setting of the Gelfand triple V ↪→ H ↪→ V ∗ with continuous and dense embeddings where H is a separable Hilbert space and V is a reflexive Banach space with dual V ∗. The system is given by dx+A(dt, x) = f(t, x)γ(dt) +B(t)u(dt), x(0...