Annals of Functional Analysis最新文献

Let (B_E) be the open unit ball of a complex finite- or infinite-dimensional Hilbert space. If f belongs to the space (mathcal {B}(B_E)) of Bloch functions on (B_E), we prove that the dilation map given by (x mapsto (1-Vert xVert ^2) mathcal {R}f(x)) for (x in B_E), where (mathcal {R}f) denotes the radial derivative of f, is Lipschitz continuous with respect to the pseudohyperbolic distance (rho _E) in (B_E), which extends to the finite- and infinite-dimensional setting the result given for the classical Bloch space (mathcal {B}). To provide this result, we will need to prove that (rho _E(zx,zy) le |z| rho _E(x,y)) for (x,y in B_E) under some conditions on (z in mathbb {C}). Lipschitz continuity of (x mapsto (1-Vert xVert ^2) mathcal {R}f(x)) will yield some applications on interpolating sequences for (mathcal {B}(B_E)) which also extends classical results from (mathcal {B}) to (mathcal {B}(B_E)). Indeed, we show that it is necessary for a sequence in (B_E) to be separated to be interpolating for (mathcal {B}(B_E)) and we also prove that any interpolating sequence for (mathcal {B}(B_E)) can be slightly perturbed and it remains interpolating.

For more than 50 years, the author has asked himself why Lorentz spaces are only defined for two parameters. Has this choice been made just for simplicity or is it a natural bound that cannot be exceeded? This question is principal and has nothing to do with usefulness. Now, I discovered a way to produce Lorentz sequence spaces for any finite number of parameters. Having found the right approach, everything turns out to be elementary; the presentation becomes an orgy of mathematical induction. Unfortunately, the new spaces are only of theoretical interest, since we do not know any handy description of their members. This dilemma is, most likely, the reason for the restriction to two, regretted above. However, by the axiom of choice, mathematicians are used to deals with objects that exist only formally; see Banach limits. Therefore, our situation is much more comfortable. It is recommended that, as a first step, readers should have a short glance at the last section, where historical aspects and the interplay between basic concepts are described. Apart from proved theorems, the paper contains many open problems. It is motivated by the same spirit as my very last bibitem in the references. Senior mathematicians should show the way into the future.

In this paper, we establish the enclosures for the spectrum of unbounded (ntimes n) operator matrices in a Banach space. For diagonally dominant and off-diagonally dominant operator matrices, we present a new Gershgorin-type results on the localization of the spectrum by using the Schur complements and the quadratic complements, respectively, that no longer requires dominance order of 0 nor (<1).

We investigate the property of commutator-simplicity in algebras from both algebraic and analytic perspectives. We demonstrate that a large class of algebras possess this property. As an analytic analog, we introduce the concept of topological commutator-simplicity for Banach algebras and establish that a (sigma )-unital (C^{*})-algebra is topological commutator-simple if and only if its multiplier algebra is. Furthermore, we explore the applications of commutator-simplicity to certain equations involving commutators, emphasizing its relevance in the study of derivations. Specifically, we obtain that every continuous local derivation on (L^1(G,omega )) is a derivation when G is a unimodular locally compact group with a diagonal bounded weight (omega ).

Let (mathcal {N}) be a non-trivial nest on a Hilbert space H and (textrm{alg}mathcal {N}) be the associated nest algebra. Let (Gin textrm{alg}mathcal {N}) be an operator with (overline{textrm{ran}(G)}in mathcal {N}backslash {H}). In this note, we give a description of Lie derivable maps and generalized Lie 2-derivable maps at G of nest algebra (textrm{alg}mathcal {N}).

Abstract

This paper deals with Fredholm properties of the one-sided coupled operator matrix ({mathcal {M}}=left( begin{array}{cc} A &{} B 0 &{} D end{array} right) left( begin{array}{cc} I &{} 0 L &{} I end{array} right)) by means of generalized Schur factorization and the associated space decompositions. For (lambda in {mathbb {C}},) some sufficient conditions are given for (lambda -{mathcal {M}}) to be Fredholm (resp. left or right Fredholm), and these conclusions are further used to determine the essential spectra of a delay equation and a wave equation with acoustic boundary conditions.