Infinite-dimensional Banach Space Research Articles

On the hereditary character of certain spectral properties and some applications

In this paper we study the behavior of certain spectral properties of an operator T on a proper closed and T-invariant subspace W ⊆ X such that Tn (X) ⊆ W, for some n ≥ 1, where T ∈ L(X) and X is an infinite-dimensional complex Banach space. We prove that for these subspaces a large number of spectral properties are transmitted from T to its restriction on W and vice-versa. As consequence of our results, we give conditions for which semiFredholm spectral properties, as well as Weyl type theorems, are equivalent for two given operators. Additionally, we give conditions under which an operator acting on a subspace can be extended on the entire space preserving the Weyl type theorems. In particular, we give some applications of these results for integral operators acting on certain functions spaces.

M-hypercyclicity of C 0-semigroup and Svep of its generator

Abstract Let 𝒯 = (Tt ) t ≥0 be a C 0-semigroup on a separable infinite dimensional Banach space X, with generator A. In this paper, we study the relationship between the single valued extension property of generator A, and the M-hypercyclicity of the C 0-semigroup. Specifically, we prove that if A does not have the single valued extension property at λ ∈ iℝ, then there exists a closed subspace M of X, such that the C 0-semigroup 𝒯 is M-hypercyclic. As a corollary, we get certain conditions of the generator A, for the C 0-semigroup to be M-hypercyclic.

On Subspace-ergodic Operators

In this paper, we define subspace-ergodic operators and give examples of these operators. We show that by any given separable infinite-dimensional Banach space, subspace-ergodic operators can be constructed. We demonstrate that an invertible operator T is subspace-ergodic if and only if T-1 is subspace-ergodic. We prove that the direct sum of two subspace-ergodic operators is subspace-ergodic and if the direct sum of two operators is subspace-ergodic, then each of them is subspace-ergodic. Also, we investigate relations between subspace-ergodic and subspace-mixing operators. For example, we show that if T is subspace-mixing and invertible, then Tn and T-n are subspace-ergodic for n∈ℕ.

On Polish groups admitting non-essentially countable actions

AbstractIt is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

Further characterizations of property (VΠ) and some applications

We carry out characterizations with techniques provided by the local spectral theory of bounded linear operators T ∈ L(X), X infinite dimensional complex Banach space, which verify property (VΠ) introduced by Sanabria et al. (Open Math. 16(1) (2018), 289-297). We also carry out the study for polaroid operators and Drazin invertible operators that verify the property mentioned above.

On the geometry of higher order Schreier spaces

For each countable ordinal α, let Sα be the Schreier set of order α and XSα be the corresponding Schreier space of order α. In this paper, we prove several new properties of these spaces. 1. If α is nonzero, then XSαpossesses the λ-property of Aron and Lohman and is a (V)-polyhedral space in the sense of Fonf and Vesely. 2. If α is nonzero and 1<p<∞, then the p-convexification XSα p possesses the uniform λ-property of Aron and Lohman. 3. For each countable ordinal α, the space XSα ∗ has the λ-property. 4. For n∈N, if T:XSn→XSnis an onto linear isometry, then Tei=±ei for each i∈N. Consequently, these spaces are light in the sense of Megrelishvili. The fact that for nonzero α, XSα is polyhedral and has the λ-property implies that each XSα is an example of a space that solves a problem of Lindenstrauss from 1966 about the existence of a polyhedral infinite dimensional Banach space whose unit ball is the closed convex hull of its extreme points. The first example of such a space was given by De Bernardi in 2017 using a renorming of c0.

Lineability, differentiable functions and special derivatives

The present work either extends or improves several results on lineability of differentiable functions and derivatives enjoying certain special properties. Among many other results, we show that there exist large algebraic structures inside the following sets of special functions: (1) The class of differentiable functions with discontinuous derivative on a set of positive measure, (2) the family of differentiable functions with a bounded, non-Riemann integrable derivative, (3) the family of functions from (0, 1) to $$\mathbb {R}$$ that are not derivatives, or (4) the family of mappings that do not satisfy Rolle’s theorem on real infinite dimensional Banach spaces. Several examples and graphics illustrate the obtained results.

Classes of Entire Analytic Functions of Unbounded Type on Banach Spaces

In this paper we investigate analytic functions of unbounded type on a complex infinite dimensional Banach space X. The main question is: under which conditions is there an analytic function of unbounded type on X such that its Taylor polynomials are in prescribed subspaces of polynomials? We obtain some sufficient conditions for a function f to be of unbounded type and show that there are various subalgebras of polynomials that support analytic functions of unbounded type. In particular, some examples of symmetric analytic functions of unbounded type are constructed.

Characterization of k-smoothness of operators defined between infinite-dimensional spaces

ABSTRACT We characterize k-smoothness of bounded linear operators defined between infinite-dimensional Hilbert spaces. We study the problem in the setting of both finite and infinite-dimensional Banach spaces. We also characterize k-smoothness of operators on some particular spaces, namely where is a finite-dimensional Banach space and is a two-dimensional Banach space. As an application, we characterize extreme contractions on where is a two-dimensional polygonal Banach space.

Remarks on retracting balls on spherical caps in \(c_{0}\), \(c\), \(l^{\infty }\) spaces

For any infinite dimensional Banach space there exists a lipschitzian retraction of the closed unit ball B onto the unit sphere S. Lipschitz constants for such retractions are, in general, only roughly estimated. The paper is illustrative. It contains remarks, illustrations and estimates concerning optimal retractions onto spherical caps for sequence spaces with the uniform norm.

Filter-dependent versions of the uniform boundedness principle

For every filter F on N, we introduce and study corresponding uniformF-boundedness principles for locally convex topological vector spaces. These principles generalise the classical uniform boundedness principles for sequences of continuous linear maps by coinciding with these principles when the filter F equals the Fréchet filter of cofinite subsets of N. We determine combinatorial properties for the filter F which ensure that these uniform F-boundedness principles hold for every Fréchet space. Furthermore, for several types of Fréchet spaces, we also isolate properties of F that are necessary for the validity of these uniform F-boundedness principles. For every infinite-dimensional Banach space X, we obtain in this way exact combinatorial characterisations of those filters F for which the corresponding uniform F-boundedness principles hold true for X.

Diametral points and diametral pairs in Banach spaces

We prove that every infinite dimensional Banach space contains symmetric, bounded, closed, convex sets with nonempty interior that have no diametral points. For diametral pairs, we show that, in a reflexive Banach space, a point in the boundary of a set of constant width C belongs to a diametral pair if and only if it is a support point of C. Therefore, if C has nonempty interior, all its boundary points belong to diametral pairs. We prove that, in some reflexive space, a set of constant width (with empty interior) does exist whose boundary contains a dense subset of points that do not belong to any diametral pair. Moreover we prove that having the whole boundary consisting of diametral pairs does not characterize constant width sets in the family of all bounded, closed, convex sets with nonempty interior of a reflexive space.

Maps preserving some spectral domains of operator products

Let X be an infinite-dimensional Banach space and let denote the algebra of all bounded linear operators on X. For the sets and are called, respectively, the semi-Fredholm domain, the Fredholm domain, and the Weyl domain of T in the spectrum, . We characterize surjective maps on (with no additivity assumption) leaving invariant or for all and .

On isomorphic embeddings of c into L1-preduals and some applications

First we study the distortion ‖T‖‖T−1‖ of an isomorphic embedding T of the space c of convergent sequences into an infinite-dimensional L1-predual X such that (extBX⁎)′⊂rBX⁎ for some r∈(0,1), that is, the set of all weak⁎ cluster points of the set of all extreme points of the closed unit ball of the dual space X⁎ is contained in the closed ball of radius r∈(0,1). We prove that ‖T‖‖T−1‖≥(3−r)/(1+r). Moreover, we give some examples showing that for every r∈(0,1) our bound is optimal. Then, we apply our theorem to provide stability results for polyhedrality and extendability of compact operators in the setting of L1-preduals. To achieve the latter goal, we additionally give a new characterization of infinite-dimensional Banach spaces having the compact norm-preserving extension property for compact operators. Furthermore, in the framework of ℓ1-preduals such that the standard basis in ℓ1 is weak⁎ convergent, we provide precise values of stability constants for the weak⁎ fixed point property.

Star-finite coverings of Banach spaces

We study star-finite coverings of infinite-dimensional normed spaces. A family of sets is called star-finite if each of its members intersects only finitely many other members of the family. It follows from our results that an LUR or a uniformly Fréchet smooth infinite-dimensional Banach space does not admit star-finite coverings by closed balls. On the other hand, we present a quite involved construction of a star-finite covering of c0(Γ) by Fréchet smooth centrally symmetric bounded convex bodies. A similar but simpler construction shows that every normed space of countable dimension (and hence incomplete) has a star-finite covering by closed balls.

Stability of phase retrievable bm$p$-frames

The phase retrieval problem presents itself in many applications in physics and engineering. Let $X$ be a Banach space and $1<p<\\infty$. Let $\\Phi=\\{x_n\\}_{n\\in~I}$ be a $p$-frame for $X$. We say $\\Phi$ is phase retrievable, if the equalities $|x^*(x_n)|=|y^*(x_n)|$ for every $n\\in~I$ imply that there exists $|\\alpha|=1$ so that $x^*=\\alpha~y^*$ for any $x^*,~y^*\\in~X^*$. In this paper, we prove that phase retrieval is always stable for $p$-frames in finite-dimensional Banach spaces and it is never uniformly stable for $p$-frames in an infinite-dimensional Banach space $X$ with a Schauder basis. We establish that stable phase retrieval is possible for the elements of $X^*$ that can be approximated sufficiently well by finite-dimensional expansion.

Strong Convergence of Mann’s Iteration Process in Banach Spaces

Mann’s iteration process for finding a fixed point of a nonexpansive mapping in a Banach space is considered. This process is known to converge weakly in some class of infinite-dimensional Banach spaces (e.g., uniformly convex Banach spaces with a Fréchet differentiable norm), but not strongly even in a Hilbert space. Strong convergence is therefore a nontrivial problem. In this paper we provide certain conditions either on the underlying space or on the mapping under investigation so as to guarantee the strong convergence of Mann’s iteration process and its variants.

Robustness of Exponential Dichotomy in a Class of Generalised Almost Periodic Linear Differential Equations in Infinite Dimensional Banach Spaces

In this paper we study the robustness of the exponential dichotomy in nonautonomous linear ordinary differential equations under integrally small perturbations in infinite dimensional Banach spaces. Some applications are obtained to the case of rapidly oscillating perturbations, with arbitrarily small periods, showing that even in this case the stability is robust. These results extend to infinite dimensions some results given in Coppel (Dichotomies in stability theory. Lecture notes in mathematics, Springer, Berlin, 1970). Based in Rodrigues (Invariância para sistemas de equacoes diferenciais com retardamento e aplicacoes, Tese de Mestrado, Universidade de Sao Paulo, Sao Carlos, 1970) and in Kloeden and Rodrigues (Nonlinear Anal 74:2695–2719, 2011), Rodrigues et al. (Stability problems in non autonomous linear differential equations in infinite dimensions. arXiv:1906.04642, 2019) we use the class of functions that we call Generalized Almost Periodic Functions that extend the usual class of almost periodic functions and are suitable to model these oscillating perturbations. We also present an infinite dimensional example of the previous results.

Equilibria in a large production economy with an infinite dimensional commodity space and price dependent preferences

We prove the existence of a competitive equilibrium in a production economy with infinitely many commodities and a measure space of agents whose preferences are price dependent. We employ a saturated measure space for the set of agents and apply recent results for an infinite dimensional separable Banach space such as Lyapunov’s convexity theorem and an exact Fatou’s lemma to obtain the result.

The isomorphic Kottman constant of a Banach space

We show that the Kottman constant $K(\cdot )$, together with its symmetric and finite variations, is continuous with respect to the Kadets metric, and they are log-convex, hence continuous, with respect to the interpolation parameter in a complex interpolation schema. Moreover, we show that $K(X)\cdot K(X^*)\geqslant 2$ for every infinite-dimensional Banach space $X$. We also consider the isomorphic Kottman constant (defined as the infimum of the Kottman constants taken over all renormings of the space) and solve the main problem left open in [Banach J. Math. Anal. 11 (2017), pp. 348–362], namely that the isomorphic Kottman constant of a twisted-sum space is the maximum of the constants of the respective summands. Consequently, the Kalton–Peck space may be renormed to have the Kottman constant arbitrarily close to $\sqrt {2}$. For other classical parameters, such as the Whitley and the James constants, we prove the continuity with respect to the Kadets metric.