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Generalized powers and measures

Burdak Zbigniew, Kosiek Marek, Pagacz Patryk, Rudol Krzysztof, Słociński Marek

Opuscula Mathematica

2021

Vol. 41, no. 6

747--754

Using the winding of measures on torus in “rational directions” special classes of unitary operators and pairs of isometries are defined. This provides nontrivial examples of generalized powers. Operators related to winding Szegö-singular measures are shown to have specific properties of their invariant subspaces.

Corona theorem for strictly pseudoconvex domains

Gwizdek Sebastian

Opuscula Mathematica

2021

Vol. 41, no. 6

843--848

Nearly 60 years have passed since Lennart Carleson gave his proof of Corona Theorem for unit disc in the complex plane. It was only recently that M. Kosiek and K. Rudol obtained the first positive result for Corona Theorem in multidimensional case. Using duality methods for uniform algebras the authors proved “abstract” Corona Theorem which allowed to solve Corona Problem for a wide class of regular domains. In this paper we expand Corona Theorem to strictly pseudoconvex domains with smooth boundaries

A simple example of a nonclosed set of representing measures

Puchała P.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

2002

Vol. 1, nr 1

197--199

Let X be a locally convex Hausdorff space, K - its compact, convex and metrizable subset. We say, that a regular Borel probability measure μ on K represents point x;ϵX if the equality f(x) = ∫fdμ holds for every f ϵ X*. We will show by a simple example, that the set of such measures supported on ext K need not be closed.