26 research outputs found
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 be a two-dimensional real normed space. In this paper we show that if
the unit circle of does not contain any line segment such that the distance
between its endpoints is greater than 1, then every transformation 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
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
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
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
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
Quantum Wasserstein isometries on the qubit state space
We describe Wasserstein isometries of the quantum bit state space with
respect to distinguished cost operators. We derive a Wigner-type result for the
cost operator involving all the Pauli matrices: in this case, the isometry
group consists of unitary or anti-unitary conjugations. In the Bloch sphere
model, this means that the isometry group coincides with the classical symmetry
group On the other hand, for the cost generated by the qubit
"clock" and "shift" operators, we discovered non-surjective and non-injective
isometries as well, beyond the regular ones. This phenomenon mirrors certain
surprising properties of the quantum Wasserstein distance.Comment: 18 pages, 3 figures. v2: new references added, v3: minor change
Wigner's theorem on Grassmann spaces
Wigner's celebrated theorem, which is particularly important in the mathematical foundations of quantum mechanics, states that every bijective transformation on the set of all rank-one projections of a complex Hilbert space which preserves the transition probability is induced by a unitary or an antiunitary operator. This vital theorem has been generalised in various ways by several scientists. In 2001, Molnár provided a natural generalisation, namely, he provided a characterisation of (not necessarily bijective) maps which act on the Grassmann space of all rank-n projections and leave the system of Jordan principal angles invariant (see [17] and [20]). In this paper we give a very natural joint generalisation of Wigner's and Molnár's theorems, namely, we prove a characterisation of all (not necessarily bijective) transformations on the Grassmann space which fix the quantity TrPQ (i.e. the sum of the squares of cosines of principal angles) for every pair of rank-n projections P and Q