Finite Dimensional Banach Spaces Research Articles

On local preservation of orthogonality and its application to isometries

We investigate the local preservation of Birkhoff-James orthogonality at a point by a linear operator on a finite-dimensional Banach space and illustrate its importance in understanding the action of the operator in terms of the geometry of the concerned spaces. In particular, it is shown that such a study is related to the preservation of k-smoothness and the extremal properties of the unit ball of a Banach space. As an application of the results obtained in this direction, we obtain a refinement of the well-known Blanco-Koldobsky-Turnsek characterization of isometries on some polyhedral Banach spaces, including ℓ∞n,ℓ1n.

Level numbers of a bounded linear operator between normed linear spaces-II

We continue the study of level numbers and level vectors of a bounded linear operator between normed linear spaces. We prove that every linear operator on a finite-dimensional polyhedral Banach space possesses only finitely many level numbers. We also explore the level numbers of weighted permutation operators between ℓ p spaces. Furthermore, we introduce the concept of approximate level numbers and study its properties and relation with level numbers. In particular, our results indicate that the concept of level numbers can be potentially applied to the spectral theory of operators, in the setting of normed linear spaces.

An approximate fixed point property

A map of a metric space into itself has the approximate fixed point property (AFPP for short) if every nearly fixed point is close to some fixed point. It is proven that both linear operators acting on finite-dimensional Banach spaces and uniformly expansive linear homeomorphisms on Banach spaces exhibit the AFPP. Furthermore, an illustration is provided of a linear homeomorphism that does not satisfy the AFPP.

On simultaneous similarity of families of commuting operators

Characterization of simultaneous similarity for commuting m m - tuples of operators is an open problem even in finite-dimensional spaces; known as “A wild problem in linear algebra”. In this paper we offer a criterion for simultaneous similarity of m m -tuples of k k -cyclic commuting operators on an arbitrary Banach space. Moreover, we obtain an additional equivalence condition in the case of finite dimensional Banach spaces, which extends the result found by Shekhtman [Math. Stat. 1 (2013), pp. 157–161] for pairs of cyclic commuting matrices. We also present two applications of our results, one in the case of general multiplication operators on Banach spaces of analytic function, and one for m m -tuples of commuting square matrices.

Some integrals for groups of bounded linear operators on finite-dimensional non-Archimedean Banach spaces

In this paper, we extend the Volkenborn integral and Shnirelman integral for groups of bounded linear operators on finite-dimensional non-Archimedean Banach spaces over $\mathbb{Q}_{p}$ and $\mathbb{C}_{p}$ respectively. When the ground field is a complete non-Archimedean valued field, which is also algebraically closed, we give some functional calculus for groups of infinitesimal generator $A$ such that $A$ is a nilpotent operator on finite-dimensional non-Archimedean Banach spaces.

Extreme contractions on finite-dimensional Banach spaces

We study extreme contractions in the setting of finite-dimensional polyhedral Banach spaces. Motivated by the famous Krein–Milman Theorem, we prove that a rank $1$ linear operator of unit norm between such spaces can be expressed as a convex combination o

Faces and Renormings of ℓ1

The faces of the unit ball of a finite-dimensional Banach space are automatically closed. The situation is different in the infinite-dimensional case. In fact, under this last condition, the closure of a face may not be a face. In this paper, we discuss these issues in an expository style. In order to illustrate the described situation we consider an equivalent renorming of the Banach space ℓ1.

Aspects of the Kahane–Salem–Zygmund inequalities in Banach spaces

The main aim of this work is to discuss several different approaches to the celebrated Kahane–Salem–Zygmund inequalities. In particular, we prove estimates for exponential Orlicz norms of averages sup _{1le jle N}Big |sum _{i=1}^K a_i(j) gamma _iBig |,, where (a_i(j)) in ell _infty ^N, , 1 le i le K and the (gamma _i) form a sequence of real or complex subgaussian random variables. Lifting these inequalities to finite dimensional Banach spaces, we get some new Kahane–Salem–Zygmund type inequalities—in particular, for spaces of subgaussian random polynomials and multilinear forms on finite dimensional Banach spaces, and also for subgaussian random Dirichlet polynomials.

Some properties of $I_3$-λ-statistical cluster points

In this note, we investigate some problems concerning the set of I_3I 3–\lambdaλ-statistical cluster points of triple sequences via ideals in finite dimensional spaces, and some of its properties in finite dimensional Banach spaces are proved.

Weighted Composition Operators between the Bloch Type Space and the Hardy Space on the Unit Ball of Complex Banach Spaces

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.

Inertial Iteration Scheme for Approximating Fixed Points of Lipschitz Pseudocontractive Maps in Arbitrary Real Banach Spaces

"We study a perturbed inertial Krasnoselskii-Mann-type algorithm and prove that the algorithm is an approximate fixed point sequence for Lipschitz pseudocontractive maps in arbitrary real Banach spaces. Strong convergence results are then established for our inertial iteration scheme for approximation of fixed points of Lipschitz pseudocontractive maps and solutions of certain important accretive-type operator equations in certain real Banach spaces. Implementation of our algorithm is illustrated using numerical examples in both finite and infinite dimensional Banach spaces. Our results improve rate of convergence and extend several related recent results. "

Regularity of the distance function from arbitrary closed sets

We investigate the distance function varvec{delta }_{K}^{phi } from an arbitrary closed subset K of a finite-dimensional Banach space (mathbf {R}^{n}, phi ) , equipped with a uniformly convex mathscr {C}^{2} -norm phi . These spaces are known as Minkowski spaces and they are one of the fundamental spaces of Finslerian geometry (see Martini et al. in Expo Math 19:97–142, 2001, https://doi.org/10.1016/S0723-0869(01)80025-6). We prove that the gradient of varvec{delta }_{K}^{phi } satisfies a Lipschitz property on the complement of the phi -cut-locus of K (a.k.a. the medial axis of mathbf {R}^{n} {{,mathrm{sim },}}K) and we prove a structural result for the set of points outside K where varvec{delta }_{K}^{phi } is pointwise twice differentiable, providing an answer to a question raised by Hiriart-Urruty (Am Math Mon 89:456–458, 1982, https://doi.org/10.2307/2321379). Our results give sharp generalisations of some classical results in the theory of distance functions and they are motivated by critical low-regularity examples for which the available results gives no meaningful or very restricted informations. The results of this paper find natural applications in the theory of partial differential equations and in convex geometry.

Multiplicative maps of matrices over rings

Let R be an arbitrary associative ring (not necessarily with an identity). A sufficient condition is given to determine when a multiplicative isomorphism on the matrix ring is additive without the assumption of idempotents. We formulate a version for matrix rings over rings with identities, and then present two applications. One is a new characterization of the automorphisms of the matrix algebra over an arbitrary field by means of multiplicative preservers, and the other one is a new characterization of topologically isomorphisms of finite dimensional real Banach spaces with dimensions no less than 2 by the existence of multiplicative isomorphisms between their rings of all bounded linear operators.

Schauder bases in Lipschitz free spaces over nets in Banach spaces

In the present note we give two explicit constructions (based on a retractional argument) of a Schauder basis for the Lipschitz free space F(N), over certain uniformly discrete metric spaces N. The first one applies to every net N in a finite dimensional Banach space, leading to the basis constant independent of the dimension. The second one applies to grids in Banach spaces with an FDD. As a corollary, we obtain a retractional Schauder basis for the Lipschitz free space F(N) over a net N in every Banach space X with a Schauder basis containing a copy of c0, as well as in every Banach space with a c0-like FDD.

Approximate orthogonality in complex normed spaces: a directional approach

We introduce and study the notion of approximate directional orthogonality in a complex normed space. We further explore the relevant scenario in the topological setting, which allows us to characterize approximate Birkhoff–James orthogonality of operators on a finite-dimensional complex Banach space. Our investigation extends some existing results in the literature to a broader scope.

$k$-smoothness on polyhedral Banach spaces

We characterize $k$-smoothness of an element on the unit sphere of a finite-dimensional polyhedral Banach space. Then we study $k$-smoothness of an operator $T \in \mathbb {L}(\ell _{\infty }^n,\mathbb {Y}),$ where $\mathbb {Y}$ is a two-dimensional Banac

Numerical radius and a notion of smoothness in the space of bounded linear operators

We observe that the classical notion of numerical radius gives rise to a notion of smoothness in the space of bounded linear operators on certain Banach spaces, whenever the numerical radius is a norm. We characterize Birkhoff-James orthogonality in the space of bounded linear operators on a finite-dimensional Banach space, endowed with the numerical radius norm. Some examples are also discussed to illustrate the geometric differences between the numerical radius norm and the usual operator norm, from the viewpoint of operator smoothness.

On the Recurrent C 0 -Semigroups, Their Existence, and Some Criteria

In this paper, recurrent C 0 -semigroups are introduced and investigated. It is proved that, despite hypercyclic C 0 -semigroups, recurrent C 0 -semigroups can be found on finite-dimensional Banach spaces. Some criteria are stated for recurrence, which is based on open sets, neighborhoods of zero, and special eigenvectors. It is established that having a dense set of recurrent vectors is a sufficient and necessary condition for a C 0 -semigroup to be recurrent. Moreover, the direct sum of recurrent C 0 -semigroups is investigated.

Generalization of the Lax–Ryabenky–Philippov theorem to nonlinear problems

In this paper, Lax’s equivalence theorem, which states that stability is a necessary and sufficient condition for its convergence in the presence of an approximation of a difference scheme, is generalized to abstract nonlinear difference problems with operators acting in finite dimensional Banach spaces. In contrast to linear finite-difference methods, such a criterion in the nonlinear case can be established only for unconditionally stable computational methods, when the corresponding a priori estimates take place for sufficiently small |h| ≤ h0. In this case, the value of h0 depends both on the consistency of discrete and continuous norms in Banach spaces, and on the magnitude of the perturbation of the input data of the problem. The proven convergence criterion is used to study the stability of difference schemes approximating quasilinear parabolic equations with nonlinearities of unbounded growth with respect to the initial data.

On the Davis-Wielandt shell of an operator and the Davis-Wielandt index of a normed linear space

We study the Davis-Wielandt shell and the Davis-Wielandt radius of an operator on a normed linear space \(\mathcal {X}\). We show that after a suitable modification, the modified Davis-Wielandt radius defines a norm on \(\mathcal {L}(\mathcal {X})\) which is equivalent to the usual operator norm on \(\mathcal {L}(\mathcal {X})\). We introduce the Davis-Wielandt index of a normed linear space and compute its value explicitly in case of some particular polyhedral Banach spaces. We also present a general method to estimate the Davis-Wielandt index of any polyhedral finite-dimensional Banach space.