Surjective linear maps preserving local spectra

Let H be an infinite dimensional complex Hilbert space and let be a surjective linear map on B(H) with (I) I 2 K(H), where K(H) denotes the closed ideal of all compact operators on H. If preserves the set of upper semi-Weyl operators and the set of all normal eigenvalues in both directions, then is an automorphism of the algebra B(H). Also the relation between the linear maps preserving the set of upper semi-Weyl operators and the linear maps preserving the set of left invertible operators is considered.

The main point of this paper is to define the multiplicity of zero and to establish Rouche's theorem for holomorphic maps in infinite dimensional complex Banach spaces.

In this paper we introduce and study the M-hypercyclicity of strongly continuous cosine function on separable complex Banach space, and we give the criteria for cosine function to be M-hypercyclic. We also prove that every separable infinite dimensional complex Banach space admits a uniformly continuous cosine function.

Let X be a finite or infinite dimensional complex Banach space. We characterize the bounded weighted composition operators between the Bloch type space and the Hardy space on the unit ball of X, extending several known results for finite dimensional domains.

We study classes of mappings between finite and infinite dimensional Banach spaces that are monotone and mappings which are differences of monotone mappings (DM). We prove a Radó–Reichelderfer estimate for monotone mappings in finite dimensional spaces that remains valid for DM mappings. This provides an alternative proof of the Fréchet differentiability a.e. of DM mappings. We establish a Morrey-type estimate for the distributional derivative of monotone mappings. We prove that a locally DM mapping between finite dimensional spaces is also globally DM. We introduce and study a new class of the so-called UDM mappings between Banach spaces, which generalizes the concept of curves of finite variation.

This paper shows if $X$ is an infinite dimensional Banach space, $X$ contains a linearly independent arc. Also based on the continuum hypothesis, that if $X$ is an infinite dimensional Banach space and card $X = c$, then $X$ contains a dense arcwise connected Hamel basis.

We prove the impossibility of expressing the space of entire functions on any infinite dimensional complex Banach space with a Schauder basis E as a countable union of spaces of entire functions that are bounded on countable open covers of E .

Julia’s lemma and Wolff’s theorem are established for (Fréchet-) holomorphic maps of bounded symmetric homogeneous domains in infinite dimensional complex Banach spaces called J ∗ {J^*} -algebras.

The angular derivative of Fréchet-holomorphic maps in J*-algebras and complex Hilbert spaces

K-POTENT PRESERVING LINEAR MAPS

Ring isomorphisms and linear or additive maps preserving zero products on nest algebras*1

SOME RESULTS RELATED TO THE SEPARABLE QUOTIENT PROBLEM

Internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. Later on, they have been generalized to the context of real vector spaces by means of the inner points. Inner points can be seen as the most opposite concept to the extreme points. In this manuscript we solve the “faceless problem”, that is, the informally posed open problem of characterizing those convex sets free of proper faces. Indeed, we prove that a non-singleton convex subset of a real vector space is free of proper faces if and only if every point of the convex set is an inner point. As a consequence, we prove the following dichotomy theorem: a point of a convex set of a real vector space is either a extreme point or an inner point of some face. We also characterize the non-singleton closed convex subsets of a topological vector space free of proper faces, which turn out to be the linear manifolds. An application of this allows us to show that the only possible minimal faces of a linearly bounded closed convex subset of a topological vector space are the extreme points. As an application of the technique we develop to prove our results, we easily construct dense proper faces of convex sets and non-dense proper faces whose closure is not a face. This fact allows us to equivalently renorm any infinite dimensional Banach space so that its unit ball contains a non-closed face. Even more, we prove that every infinite dimensional Banach space containing an isomorphic copy of $$c_0$$ or $$\ell _p$$ , $$1<p<\infty $$ , can be equivalently renormed so that its unit ball contains a face whose closure is not a face.

This paper consists of two main results. In the first one we describe all Kaehler immersions of a bounded symmetric domain into the infinite dimensional complex projective space in terms of the Wallach set of the domain. In the second one we exhibit an example of complete and non-homogeneous Kaehler-Einstein metric with negative scalar curvature which admits a Kaehler immersion into the infinite dimensional complex projective space.

Two homogeneous measures of noncompactness β and γ on an infinite dimensional Banach space X are called “equivalent” if there exist positive constants b and c such that bβ(S) ≤ γ(S) ≤ cβ(S) for all bounded sets \({S\subset X}\). If such constants do not exist, the measures of noncompactness are “inequivalent.” We ask a foundational question which apparently has not previously been considered: For what infinite dimensional Banach spaces do there exist inequivalent measures of noncompactness on X? We provide here the first examples of inequivalent measures of noncompactness. We prove that such inequivalent measures exist if X is a Hilbert space; or if (Ω, Σ, μ) is a general measure space, 1 ≤ p ≤ ∞, and X = Lp(Ω, Σ, μ); or if K is a compact Hausdorff space and X = C(K); or if K is a compact metric space, 0 < λ ≤ 1, and X = C0,λ(K), the Banach space of Holder continuous functions with Holder exponent λ. We also prove the existence of such inequivalent measures of noncompactness if Ω is an open subset of \({\mathbb{R}^n}\) and X is the Sobolev space Wm,p(Ω). Our motivation comes from questions about existence of eigenvectors of homogeneous, continuous, order-preserving cone maps f : C→C and from the closely related issue of giving the proper definition of the “cone essential spectral radius” of such maps. These questions are considered in the companion paper [28]; see, also, [27].