Search

The Sz\'asz inequality is a classical result providing a bound for polynomials with zeros in the upper half of the complex plane in terms of its low-order coefficients. Some generalisations of this result to polynomials in several variables were done by Borcea-Br\"and\'en and Knese. In this article the inequality is discussed in the context of matrix polynomials and matrix variables. As a byproduct we improve the scalar Sz\'asz inequality by relaxing the assumption of the location of zeros.

In a paper from 2012 Jab{\l}o\'nski, Jung and Stochel introduced the weighted shifts on directed trees, a generalisation of well known weighted shift operators on $\ell^2$. In the last decade this class has proven itself handy for finding counterexamples in operator theory. Properties of underlying graph structure had essential influence on the operator. It appears that a slight generalisation of the class, namely weighted shifts on directed forests, shows even deeper relations between graph theory and functional analysis. Several operations on directed forests have their natural operator-theoretic counterparts. This paper is meant to present advantages of the directed forest approach. As an application of the interrelation between graphs and operators we provide full characterisation of directed forests on which every hyponormal bounded weighted shift is power hyponormal., Comment: 16 pages, 7 figures

Weighted shifts on directed trees are a decade old generalisation of classical shift operators in the sequence space $\ell^2$. In this paper we introduce the joint backward extension property (JBEP) for classes of weighted shifts on directed trees. If a class satisfies JBEP, the existence of a common backward extension within the class for a family of weighted shifts on rooted directed trees does not depend on the additional structure of the big tree (of fixed depth). We decide whether several classes of operators have JBEP. For subnormal or power hyponormal weighted shifts the property is satisfied, while it fails for completely hyperexpansive or quasinormal. Nevertheless some positive results on joint backward extensions of completely hyperexpansive weighted shifts are proven., Comment: 33 pages, 6 figures. Continuation of arXiv:2109.05970

The weighted shifts are long known and important class of operators. One of known generalisation of this class are weighted shifts on directed trees, where we replace the linear order of coordinates in $\ell^2$ with a possibly more sophisticated graph structure. In this paper we focus on the question whether a weighted shift on a directed tree admits a subnormal or just power hyponormal (i.e. all powers of the operator are hyponormal) backward extension (a shift on larger directed tree). It comes out that in both cases the question whether we can obtain "joint extension" for a family of trees does not depend on any deep interrelations between the given trees but on their own "extendability" only. We introduce a generalised framework of weighted shifts on directed forests which seems to be slightly more convenient to work with. The characterisation of all the leafless directed forests on which all hyponormal weighted shifts are power hyponormal is also given.

While topology given by a linear order has been extensively studied, this cannot be said about the case when the order is given only locally. The aim of this paper is to fill this gap. We consider relation between local orderability and separation axioms and give characterisation of all connected, locally connected or compact locally ordered Hausdorff spaces. A collection of interesting examples is also offered.

A soft presentation of hyperbolic spaces, free of differential apparatus, is offered. Fifth Euclid's postulate in such spaces is overthrown and, among other things, it is proved that spheres (equipped with great-circle distances) and hyperbolic and Euclidean spaces are the only locally compact geodesic (i.e., convex) metric spaces that are three-point homogeneous., Comment: 14 pages