Marek Kosiek

Reflexivity of N-tuples of contractions with rich joint left essential spectrum

It will be shown that WOT-closed algebra generated by N-tuple of double commuting contractions, for which the polydiscDN is a spectral set and whose joint left essential spectrum is dominating for the algebraH∞(DN) is reflexive. The second version of our main result, instead of double commutativity, uses the membership of the classC0.

On the reflexivity of pairs of contractions

We consider pairs of commuting contractions such that the joint left essential spectrum is dominating for the algebra H°°(D2). It is also assumed, in the first case, that one of them is C0, and the second one is absolutely continuous. In the second case, we assume that the pair is diagonally extendable. It will be shown that such pairs are reflexive.

Fast Convergence of Distance-based Inconsistency in Pairwise Comparisons

This study presents theoretical proof and empirical evidence of the reduction algorithm convergence for the distance-based inconsistency in pairwise comparisons. Our empirical research shows that the convergence very quick. It usually takes less than 10 reductions to bring the inconsistency of the pairwise comparisons matrix below the assumed threshold of 1/3 sufficient for most applications. We believe that this is the first Monte Carlo study demonstrating such results for the convergence speed of inconsistency reduction in pairwise comparisons.

Dual Algebras and A-Measures

Weak-star closures of Gleason parts in the spectrum of a function algebra are studied. These closures relate to the bidual algebra and turn out both closed and open subsets of a compact hyperstonean space. Moreover, weak-star closures of the corresponding bands of measures are reducing. Among the applications we have a complete solution of an abstract version of the problem, whether the set of nonnegative A-measures (called also Henkin measures) is closed with respect to the absolute continuity. When applied to the classical case of analytic functions on a domain of holomorphy , our approach avoids the use of integral formulae for analytic functions, strict pseudoconvexity, or some other regularity of . We also investigate the relation between the algebra of bounded holomorphic functions on and its abstract counterpart—the * closure of a function algebra A in the dual of the band of measures generated by one of Gleason parts of the spectrum of A.

On common invariant subspaces for commuting contractions with rich spectrum (Indiana Univ. Math. J. \textbf{53} (2004), 823--844): erratum

An error was found in the proofs of Theorems 4.4 and 4.5 of [3]. This paper replaces said proofs by weaker results, which are, however, strong enough to prove all other theorems of [3], in particular all the results concerning invariant subspaces.

Measures orthogonal to tensor products of function algebras

Some fundamental properties of measures orthogonal to tensor products of function algebras are considered. It is shown that if M 1 , M 2 are reducing bands of measures for algebras A 1 , A 2 , then their projection-preserving product is a reducing band for A 1 ⊗ A 2 . As an application, a decomposition of measures orthogonal to the tensor product of some classes of function algebras is obtained. These classes contain among others, multidimensional ball algebras.