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Dragoljub J. Kečkić

Publications and source records attributed to Dragoljub J. Kečkić.

12 recordsLinked to original sources
arXiv2025

Clarkson-McCarthy inequality on a locally compact group

Let $G$ be a locally compact group, $μ$ its Haar measure, $\hat G$ its Pontryagin dual and $ν$ the dual measure. For any $A_θ\in L^1(G;\mathcal C_p)\cap L^2(G;\mathcal C_p)$, ($\mathcal C_p$ is Schatten ideal), and $1<p\le2$ we prove $$\int_{\hat G}\left\|\int_GA_θ\overline{ξ(θ)}\,\mathrm dμ(θ)\right\|_p^q\,\mathrm dν(ξ)\le \left(\int_G\|A_θ\|_p^p\,\mathrm dμ(θ)\right)^{q/p}, $$ where $q=p/(p-1)$. This appears to be a generalization of some earlier obtained inequalities, including Clarkson-McCarthy inequalities (in the case $G=\mathbf Z_2$), and Hausdorff-Young inequality. Some corollaries are also given.

math.FA↗
arXiv2024

Measures of noncompactness in Hilbert $C^*$-modules

Consider a countably generated Hilbert $C^*$-module $\mathcal M$ over a $C^*$-algebra $\mathcal A$. There is a measure of noncompactness $λ$ defined, roughly as the distance from finitely generated projective submodules, which is independent of any topology. We compare $λ$ to the Hausdorff measure of noncompactness with respect to the family of seminorms that induce a topology recently iontroduced by Troitsky, denoted by $χ^*$. We obtain $λ\equivχ^*$. Related inequalities involving other known measures of noncompactness, e.g. Kuratowski and Istrăţescu are laso obtained as well as some related results on adjontable operators.

math.OA↗
arXiv2023

Isolated vertices and diameter of the $BJ$-orthograph in $C^*$-algebras

We give necessary and sufficient condition that an element of an arbitrary $C^{*}$-algebra is an isolated vertex of the orthograph related to the mutual strong Birkhoff-James orthogonality. Also, we prove that for all $C^{*}$-algebras except $\mathbb{C},\mathbb{C}\oplus\mathbb{C}$ and $M_2(\mathbb{C})$ all non isolated points make a single connected component of the orthograph which diameter is less than or equal to $4$, i.e. any two non isolated points can be connected by a path with at most $4$ edges. Some related results are given.

math.OA↗
arXiv2018

On the continuity of probabilistic distance

The famous result of B.~Schweizer and A.~Sklar [Pacific J Math 10(1960) 313--334 - Theorem 8.2] asserts that, given a probabilistic metric space $(X,\mathcal F,t)$, $\mathcal F=\{F_{p,q}:p,q\in X\}$, we have $F_{p_n,q_n}(x)\to F_{p,q}(x)$ provided that $F_{p,q}$ is continuous at $x$ and $t$ is continuous and stronger then Łukasiwicz's $t$-norm. We extend this result to arbitrary continuous triangular norms, i.e.\ we omit the condition "$t$ is stronger then Łukasiewicz's".

math.FA↗
arXiv2018

Continuous generalization of Clarkson-McCarthy inequalities

Let $G$ be a compact abelian group, let $μ$ be the corresponding Haar measure, and let $\hat G$ be the Pontryagin dual of $G$. Further, let $C_p$ denote the Schatten class of operators on some separable infinite dimensional Hilbert space, and let $L^p(G;C_p)$ denote the corresponding Bochner space. If $G\niθ\mapsto A_θ$ is the mapping belonging to $L^p(G;C_p)$ then, $$\sum_{k\in\hat G}\left\|\int_G\overline{k(θ)}A_θ\,\mathrm{d}θ\right\|_p^p\le\int_G\|A_θ\|_p^p\,\mathrm{d}θ,\qquad p\ge2$$ $$\sum_{k\in\hat G}\left\|\int_G\overline{k(θ)}A_θ\,\mathrm{d}θ\right\|_p^p\le\left(\int_G\|A_θ\|_p^q\,\mathrm{d}θ\right)^{p/q},\qquad p\ge2.$$ $$\sum_{k\in\hat G}\left\|\int_G\overline{k(θ)}A_θ\,\mathrm{d}θ\right\|_p^q\le\left(\int_G\|A_θ\|_p^p\,\mathrm{d}θ\right)^{q/p},\qquad p\le2.$$ If $G$ is a finite group, the previous comprises several earlier obtained generalizations of Clarkson-McCarthy inequalities (e.g. $G=\mathbf{Z}_n$ or $G=\mathbf{Z}_2^n$), as well as the original inequalities, for $G=\mathbf{Z}_2$. Other related inequalities are also obtained.

math.FA↗
arXiv2017

Measures of noncompactness on the standard Hilbert $C^*$-module

We define a measure of noncompactness $λ$ on the standard Hilbert $C^*$-module $l^2(\mathcal A)$ over a unital $C^*$-algebra, such that $λ(E)=0$ if and only if $E$ is $\mathcal A$-precompact (i.e.\ it is $\varepsilon$-close to a finitely generated projective submodule for any $\varepsilon>0$) and derive its properties. Further, we consider the known, Kuratowski, Hausdorff and Istrăţescu measure of noncomapctnes on $l^2(\mathcal A)$ regarded as a locally convex space with respect to a suitable topology, and obtain their properties as well as some relationship between them and introduced measure of noncompactness $λ$.

math.OA↗
arXiv2015

Fredholm operators on $C^*$-algebras

The aim of this note is to generalize the notion of Fredholm operator to an arbitrary $C^*$-algebra. Namely, we define "finite type" elements in an axiomatic way, and also we define Fredholm type element $a$ as such element of a given $C^*$-algebra for which there are finite type elements $p$ and $q$ such that $(1-q)a(1-p)$ is "invertible". We derive index theorem for such operators. In applications we show that classical Fredholm operators on a Hilbert space, Fredholm operators in the sense of Breuer, Atiyah and Singer on a properly infinite von Neumann algebra, and Fredholm operators on Hilbert $C^*$-modules over an unital $C^*$-algebra in the sense of Mishchenko and Fomenko are special cases of our theory.

math.OA↗
arXiv2015

The index of a subspatial product system over a Hilbert $C^*$-module -- an example

Recently, Bhat, Liebscher and Skeide exhibited an unusual example of product system of Hilbert $C^*$ modules. Namely, a subsystem of a Fock system, that is not Fock. Later, the authors defined the index of any product system of Hilbert $C^*$-modules, generalizing earlier partial definitions. In this note we compute the index of the mentioned interesting subsystem, and also we determine all its units.

math.OA↗
arXiv2012

TVS-cone metric spaces as a special case of metric spaces

There have been a number of generalizations of fixed point results to the so called TVS-cone metric spaces, based on a distance function that takes values in some cone with nonempty interior (solid cone) in some topological vector space. In this paper we prove that the TVS-cone metric space can be equipped with a family of mutually equivalent (usual) metrics such that the convergence (resp. property of being Cauchy sequence, contractivity condition) in TVS sense is equivalent to convergence (resp. property of being Cauchy sequence, contractivity condition) in all of these metrics. As a consequence, we prove that if a topological vector space $E$ and a solid cone $P$ are given, then the category of TVS-cone metric spaces is a proper subcategory of metric spaces with a family of mutually equivalent metrics (Corollary 3.9). Hence, generalization of a result from metric spaces to TVS-cone metric spaces is meaningless. This, also, leads to a formal deriving of fixed point results from metric spaces to TVS-cone metric spaces and makes some earlier results vague. We also give a new common fixed point result in (usual) metric spaces context, and show that it can be reformulated to TVS-cone metric spaces context very easy, despite of the fact that formal (syntactic) generalization is impossible. Apart of main results, we prove that the existence of a solid cone ensures that the initial topology is Hausdorff, as well as it admits a plenty of convex open sets. In fact such topology is stronger then some norm topology.

math.FA↗
arXiv2011

On the index of product systems of Hilbert modules

In this note we prove that the set of all uniformly continuous units on a product system over a C* algebra B can be endowed with the structure of left right B - B Hilbert module after identifying similar units by the suitable equivalence relation. We use this construction to define the index of the initial product system, and prove that it is the generalization of earlier defined indices by Arveson (in the case B=C) and Skeide (in the case of spatial product system). We prove that such defined index is a covariant functor from the category od continuous product systems to the category of B bimodules. We also prove that the index is subadditive with respect to the outer tensor product of product systems, and prove additional properties of the index of product systems that can be embedded into a spatial one.

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