Asymptotic Behaviour and Cyclic Properties of Weighted Shifts on Directed Trees

In this paper we investigate a new class of operators called weighted shifts on directed trees introduced recently in [Z. J. Jablonski, I. B. Jung and J. Stochel, A Non-hyponormal Operator Generating Stieltjes Moment Sequences, J. Funct. Anal. 262 (2012), no. 9, 3946--3980.]. This class is a natural generalization of the so called weighted bilateral, unilateral and backward shift operators. In the first part of the paper we calculate the asymptotic limit and the isometric asymptote of a contractive weighted shift on a directed tree and that of the adjoint. Then we use the asymptotic behaviour and similarity properties to deal with cyclicity. We also show that a weighted backward shift operator is cyclic if and only if there is at most one zero weight.Comment: 22 page

A contribution to the Aleksandrov conservative distance problem in two dimensions

Let $E$ be a two-dimensional real normed space. In this paper we show that if the unit circle of $E$ does not contain any line segment such that the distance between its endpoints is greater than 1, then every transformation $\phi\colon E\to E$ which preserves the unit distance is automatically an affine isometry. In particular, this condition is satisfied when the norm is strictly convex.Comment: 8 pages, 3 figure

Maps on classes of Hilbert space operators preserving measure of commutativity

In this paper first we give a partial answer to a question of L. Moln\'ar and W. Timmermann. Namely, we will describe those linear (not necessarily bijective) transformations on the set of self-adjoint matrices which preserve a unitarily invariant norm of the commutator. After that we will characterize those (not necessarily linear or bijective) maps on the set of self-adjoint rank-one projections acting on a two-dimensional complex Hilbert space which leave the latter quantity invariant. Finally, this result will be applied in order to obtain a description of such bijective preservers on the unitary group and on the set of density operators.Comment: 16 pages, submitted to a journa

Asymptotic behaviour of Hilbert space operators with applications

This dissertation summarizes my investigations in operator theory during my PhD studies. The first chapter is an introduction to that field of operator theory which was developed by B. Sz.-Nagy and C. Foias, the theory of power-bounded Hilbert space operators. In the second and third chapter I characterize operators which arise from power-bounded operators asymptotically. Chapter 4 is devoted to provide a possible generalization of (the necessity part of) Sz.-Nagy's famous similarity theorem. In Chapter 5 I collected my results concerning the commutant mapping of asymptotically non-vanishing contractions. In the final chapter the reader can find results about cyclic properties of weighted shift operators on directed trees.Comment: 96 pages, 6 chapters, 3 figures. Page 87-89 was written in Hungarian, but it is the same as page 84-86. phd thesis, University of Szege

Surjective Lévy-Prokhorov Isometries

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\'evy-Prokhorov distance. Motivated by these facts and some recent works related to the characterisation of onto isometries of spaces of Borel probability measures, here we give the complete description of the structure of surjective L\'evy-Prokhorov isometries on the space of all Borel probability measures on an arbitrary separable real Banach space. Our result can be considered as a generalisation of L. Moln\'ar's earlier result which characterises surjective L\'evy isometries of the space of all probability distribution functions on the real line. However, the present more general setting requires the development of an essentially new technique

The structure of maps on the space of all quantum pure states that preserve a fixed quantum angle

Abstract Let $H$ be a Hilbert space and $P(H)$ be the projective space of all quantum pure states. Wigner’s theorem states that every bijection $\phi \colon P(H)\to P(H)$ that preserves the quantum angle between pure states is automatically induced by either a unitary or an antiunitary operator $U\colon H\to H$. Uhlhorn’s theorem generalizes this result for bijective maps $\phi$ that are only assumed to preserve the quantum angle $\frac{\pi }{2}$ (orthogonality) in both directions. Recently, two papers, written by Li–Plevnik–Šemrl and Gehér, solved the corresponding structural problem for bijections that preserve only one fixed quantum angle $\alpha$ in both directions, provided that 0 < \alpha \leq \frac{\pi }{4} holds. In this paper we solve the remaining structural problem for quantum angles $\alpha$ that satisfy \frac{\pi }{4} < \alpha < \frac{\pi }{2}, hence complete a programme started by Uhlhorn. In particular, it turns out that these maps are always induced by unitary or antiunitary operators, however, our assumption is much weaker than Wigner’s

Isometric study of Wasserstein spaces - the real line

Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space $\mathcal {W}_2(\mathbb{R}^n)$. It turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow. Following this line of investigation, we compute $\mathrm {Isom}(\mathcal {W}_p(\mathbb{R}))$, the isometry group of the Wasserstein space $\mathcal {W}_p(\mathbb{R})$ for all $p \in [1, \infty )\setminus \{2\}$. We show that $\mathcal {W}_2(\mathbb{R})$ is also exceptional regarding the parameter $p$: $\mathcal {W}_p(\mathbb{R})$ is isometrically rigid if and only if $p\neq 2$. Regarding the underlying space, we prove that the exceptionality of $p=2$ disappears if we replace $\mathbb{R}$ by the compact interval $[0,1]$. Surprisingly, in that case, $\mathcal {W}_p([0,1])$ is isometrically rigid if and only if $p\neq 1$. Moreover, $\mathcal {W}_1([0,1])$ admits isometries that split mass, and $\mathrm {Isom}(\mathcal {W}_1([0,1]))$ cannot be embedded into $\mathrm {Isom}(\mathcal {W}_1(\mathbb{R}))$

On isometries of Wasserstein spaces (Research on preserver problems on Banach algebras and related topics)

It is known that if p ≥ 1, then the isometry group of the metric space (X, ϱ) embeds into the isometry group of the Wasserstein space Wp(X, ϱ). Those isometries that belong to the image of this embedding are called trivial. In many concrete cases, all isometries are trivial, however, this is not always the case. The aim of this survey paper is to provide a structured overview of recent results concerning trivial and different types of nontrivial isometries