The article surveys a number of potential theory results in the discrete setting of trees and in an application to complex analysis. On trees for which the associated random walk is recurrent, we discuss Riesz decomposition, flux, a type of potential called H-potential, and present a new result dealing with the boundary behaviour of H-potentials on a specific recurrent homogeneous tree. On gene...

The invertible, compact and Fredholm multiplication operators on generalized Lorentz-Zygmund (GLZ) spaces Lp,q;α, 1 < p ≤ ∞, 1 ≤ q ≤ ∞, α in the Euclidean space R, are characterized in this paper.

We use elementary algebraic properties of left, right multiplication operators to prove some deep structural left $m$-invertible, $m$-isometric, $m$-selfadjoint and other related classes Banach space operators, often adding value extant results.

Recently, the utilization of invariant aggregation operators, i.e., aggregation operators not depending on a given scale of measurement was found as a very current theme. One type of invariantness of aggregation operators is the homogeneity what means that an aggregation operator is invariant with respect to multiplication by a constant. We present here a complete characterization of homogeneou...

Let (Ω,Σ,μ) be a σ -finite complete measure space and C be the field of complex numbers. By L(μ ,CN), we denote the linear space of all equivalence classes of CN-valued Σ-measurable functions on Ω that are identified μ-a.e. and are considered as column vectors. Let M◦ denote the linear space of all functions in L(μ ,CN) that are finite a.e. With the topology of convergence in measure on the set...

Let S be the multiplication operator by an independent variable x in L2(0, 1) and V be an integral operator of Volterra type. We find conditions for T := S + V to be similar to S and discuss some generalisations of the results obtained to an abstract setting.