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Tamás Titkos

Publications and source records attributed to Tamás Titkos.

29 records · Page 2Linked to original sources
arXiv2019

On isometric embeddings of Wasserstein spaces -- the discrete case

The aim of this short paper is to offer a complete characterization of all (not necessarily surjective) isometric embeddings of the Wasserstein space $\mathcal{W}_p(\mathcal{X})$, where $\mathcal{X}$ is a countable discrete metric space and $0<p<\infty$ is any parameter value. Roughly speaking, we will prove that any isometric embedding can be described by a special kind of $\mathcal{X}\times(0,1]$-indexed family of nonnegative finite measures. Our result implies that a typical non-surjective isometric embedding of $\mathcal{W}_p(\mathcal{X})$ splits mass and does not preserve the shape of measures. In order to stress that the lack of surjectivity is what makes things challenging, we will prove alternatively that $\mathcal{W}_p(\mathcal{X})$ is isometrically rigid for all $0<p<\infty$.

math.FA↗
arXiv2018

Operators on anti-dual pairs: Generalized Krein--von Neumann extension

The main aim of this paper is to generalize the classical concept of positive operator, and to develop a general extension theory, which overcomes not only the lack of a Hilbert space structure, but also the lack of a normable topology. The concept of anti-duality carries an adequate structure to define positivity in a natural way, and is still general enough to cover numerous important areas where the Hilbert space theory cannot be applied. Our running example -- illustrating the applicability of the general setting to spaces bearing poor geometrical features -- comes from noncommutative integration theory. Namely, representable extension of linear functionals of involutive algebras will be governed by their induced operators. The main theorem, to which the vast majority of the results is built, gives a complete and constructive characterization of those operators that admit a continuous positive extension to the whole space. Various properties such as commutation, or minimality and maximality of special extensions will be studied in detail.

math.FA↗
arXiv2017

Quasi-units as orthogonal projections

The notion of quasi-unit has been introduced by Yosida in unital Riesz spaces. Later on, a fruitful potential theoretic generalization was obtained by Arsove and Leutwiler. Due to the work of Eriksson and Leutwiler, this notion also turned out to be an effective tool by investigating the extreme structure of operator segments. This paper has multiple purposes which are interwoven, and are intended to be equally important. On the one hand, we identify quasi-units as orthogonal projections acting on an appropriate auxiliary Hilbert space. As projections form a lattice and are extremal points of the effect algebra, we conclude the same properties for quasi-units. Our second aim is to apply these results for nonnegative sesquilinear forms. Constructing an order preserving bijection between operator- and form segments, we provide a characterization of being extremal in the convexity sense, and we give a necessary and sufficient condition for the existence of the greatest lower bound of two forms. Closing the paper we revisit some statements by using the machinery developed by Hassi, Sebestyén, and de Snoo. It will turn out that quasi-units are exactly the closed elements with respect to the antitone Galois connection induced by parallel addition and subtraction.

math.FA↗
arXiv2017

On the uniqueness of the Lebesgue decomposition of normal states on $B(H)$

The non-commutative theory of the Lebesgue-type decomposition of positive functionals is originated with S. P. Gudder. Although H. Kosaki's counterexample shows that the decomposition is not unique in general, the complete characterization of uniqueness is still not known. Using the famous operator-decomposition of T. Ando, we give a necessary and sufficient condition for uniqueness in the particular case when the underlying algebra is $B(H)$, the $C^*$-algebra of all continuous linear operators on a Hilbert space $H$. Namely, given a normal state $f$, the $f$-Lebesgue decomposition of any other normal state is unique if and only if the representing trace class operator of $f$ has finite rank. Some recent results tell that the decomposition is unique over a large class of commutative algebras. Our characterization demonstrates that the lack of commutativity is not the real cause of non-uniqueness.

math.OA↗
arXiv2017

A characterization of positive normal functionals on the full operator algebra

Using the recent theory of Krein--von Neumann extensions for positive functionals we present several simple criteria to decide whether a given positive functional on the full operator algebra is normal. We also characterize those functionals defined on the left ideal of finite rank operators that have a normal extension.

math.FA↗
arXiv2017

A characterisation of isometries with respect to the Lévy-Prokhorov metric

According to the fundamental work of Yu.V. Prokhorov, the general theory of stochastic processes can be regarded as the theory of probability measures in complete separable metric spaces. Since stochastic processes depending upon a continuous parameter are basically probability measures on certain subspaces of the space of all functions of a real variable, a particularly important case of this theory is when the underlying metric space has a linear structure. Prokhorov also provided a concrete metrisation of the topology of weak convergence today known as the L{é}vy-Prokhorov distance. Motivated by these facts, the famous Banach-Stone theorem, and some recent works related to characterisations of onto isometries of spaces of Borel probability measures, here we give a complete description of surjective isometries with respect to the L{é}vy-Prokhorov metric in case when the underlying metric space is a separable Banach space. Our result can be considered as a generalisation of L. Molnár's earlier Banach-Stone-type result which characterises onto isometries of the space of all probability distribution functions on the real line wit respect to the Lévy distance. However, the present more general setting requires the development of an essentially new technique.

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arXiv2016

Arlinskii's iteration and its applications

Several Lebesgue-type decomposition theorems in analysis have a strong relation to the operation called: parallel sum. The aim of this paper is to investigate this relation from a new point of view. Namely, using a natural generalization of Arlinskii's approach (which identifies the singular part as a fixed point of a single-variable map) we prove the existence of a Lebesgue-type decomposition for nonnegative sesquilinear forms. As applications, we also show that how this approach can be used to derive analogous results for representable functionals, nonnegative finitely additive measures, and positive definite operator functions. The focus is on the fact that each theorem can be proved with the same completely elementary method.

math.FA↗
arXiv2016

On the order structure of representable functionals

The main purpose of this paper is to investigate some natural problems regarding the order structure of representable functionals on $^*$-algebras. We describe the extreme points of order intervals, and give a nontrivial sufficient condition to decide whether or not the infimum of two representable functionals exists. To this aim we offer a suitable approach to the Lebesgue decomposition theory, which is in complete analogy with the one developed by Ando in the context of positive operators. This tight analogy allows to invoke Ando's results to characterize uniqueness of the decomposition, and solve the infimum problem over certain operator algebras.

math.FA↗
arXiv2014

Positive Definite Operator Functions and Sesquilinear Forms

Due to the fundamental works of T. Ando, W. Szymański, F. H. Szafraniec, and many others it is well known that sesquilinear forms play an important role in dilation theory. The crucial fact is that every positive definite operator function induces a sesquilinear form in a natural way. The purpose of this survey-like paper is to apply some recent results of Z. Sebestyén, Zs. Tarcsay, and the author for such functions. While most of the results are not new, the paper's main contribution is the unified discussion from the viewpoint of sesquilinear forms.

math.FA↗
arXiv2014

A Short-type Decomposition Of Forms

The main purpose of this paper is to present a decomposition theorem for nonnegative sesquilinear forms. The key notion is the short of a form to a linear subspace. This is a generalization of the well-known operator short defined by M. G. Krein. A decomposition of a form into a shorted part and a singular part (with respect to an other form) will be called short-type decomposition. As applications, we present some analogous results for bounded positive operators acting on a Hilbert space; for additive set functions on a ring of sets; and for representable positive functionals on a $\ast$-algebra.

math.FA↗