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Marek Kosiek

Publications and source records attributed to Marek Kosiek.

10 recordsLinked to original sources
arXiv2026

General solution of corona problem

Using a description of the spectrum of bidual algebra $A^{**}$ of a uniform algebra $A$ we obtain abstract corona theorem for certain uniform algebras. It asserts the density of a specific Gleason part in the spectrum of an $H^\infty$ -- type subalgebra of $A^{**}$. There is an isometric isomorphism of the latter subalgebra with $H^\infty(G)$ for a wide class of domains $G\subset\mathbb C^d$. Using abstract corona theorem we show the density of the canonical image of $G$ in the spectrum of $H^\infty(G)$, solving positively corona problem for such domains. In particular, we obtain positive solution for balls and polydisks.

math.FA↗
arXiv2026

Spectrum of bidual uniform algebras

We obtain a description of the spectrum of bidual algebra $A^{**}$ of a uniform algebra $A$. This spectrum turns out to be a quotient space of the hyper-Stonean envelope of the spectrum of $A$.

math.FA↗
arXiv2025

Corona Theorem

For a wide class of domains $G\subset\mathbb C^d$ including balls and polydisks we prove the density of their canonical image in the spectrum of $H^\infty(G)$. This Corona Theorem is proved first in its abstract version for certain uniform algebras. We use properties of bands of measures and idempotents corresponding to Gleason parts. The essential tools are properties of hyper-Stonean spaces, normal and Henkin measures and some ideas based on Hoffman - Rossi theorem. We also use our previous results on weak-star closures of reducing bands of measures. Uniform bounds for operators used to solve the Gleason problem concerning ideals of analytic functions vanishing at given points are applied for bidual algebras.

math.FA↗
arXiv2024

An operator theory approach to the evanescent part of a two-parametric weak-stationary stochastic process

A new approach to the evanescent part of a two-dimensional weak-stationary stochastic process with the past given by a half-plane is proceed. The classical result due to Helson and Lowdenslager divides a two-parametric weak-stationary stochastic process into three parts. In this paper we describe the most untouchable one - the evanescent part. Moreover, we point out how this part depends on the shape of the past.

math.PR↗
arXiv2023

A disintegration theorem

A new approach to disintegration of measures is presented, allowing one to drop the usually taken separability assumption. The main tool is a result on fibers in the spectrum of algebra of essentially bounded functions established recently by the first-named author.

math.FA↗
arXiv2023

Gleason parts of bidual algebras

It is shown that the embeding of any Gleason part of a uniform algebra into the spectrum of its second dual is an entire Gleason part. This result is based on the equality of weak-star and norm topologies on the Bear-Gleason part.

math.FA↗
arXiv2019

The decomposition theorems in Baer $*$-rings

We show a general decomposition theorem in Baer *-rings. As a consequence the vast majority of decompositions known in the algebra of bounded Hilbert space operators are generalized to Baer *-rings. There are also results which are new in the algebra of bounded Hilbert space operators. The model of summands in Wold-Słociński decomposition in Baer *-rings is given.

math.RA↗
arXiv2012

On the unitary part of isometries in commuting, completely non doubly commuting pairs

There are considered isometries on a Hilbert space. By the Wold theorem any isometry can be decomposed into a unitary operator and a unilateral shift. For a pair of isometries, even commuting, a maximal subspace reducing one isometry to a unitary operator might not reduce the other isometry. In the paper are considered pairs of commuting isometries which are completely non doubly commuting. For such pairs there are no nontrivial subspaces reducing both isometries and one of them to a unitary operator. The results describe a unitary part of an isometry in such a pair.

math.FA↗
arXiv2011

Fibers of $L^{\infty}$ algebra

It is shown that Gelfand transforms of elements $f\in\lmu$ are constant at almost every fiber $Π^{-1}(\{x\})$ of the spectrum of $\lmu$ in the following sense: for each $f\in\lmu$ there is an open dense subset $U=U(f)$ of this spectrum having full measure and such that the Gelfand transform of $f$ is constant on the intersection $Π^{-1}(\{x\})\cap U$.

math.FA↗