Dmitry V. Yakubovich

Computation of conformal representations of compact Riemann surfaces

We find a system of two polynomial equations in two unknowns, whose solution allows us to give an explicit expression of the conformal representation of a simply connected three-sheeted compact Riemann surface onto the extended complex plane. This function appears in the description of the ratio asymptotic of multiple orthogonal polynomials with respect to so-called Nikishin systems of two measures.

Følner sequences and finite operators

Abstract This article analyzes Folner sequences of projections for bounded linear operators and their relationship to the class of finite operators introduced by Williams in the 70s. We prove that each essentially hyponormal operator has a proper Folner sequence (i.e., an increasing Folner sequence of projections strongly converging to 1 ). In particular, any quasinormal, any subnormal, any hyponormal and any essentially normal operator has a proper Folner sequence. Moreover, we show that an operator is finite if and only if it has a proper Folner sequence or if it has a non-trivial finite dimensional reducing subspace. We also analyze the structure of operators which have no Folner sequence and give examples of them. For this analysis we introduce the notion of strongly non-Folner operators, which are far from finite block reducible operators, in some uniform sense, and show that this class coincides with the class of non finite operators.

Følner Sequences in Operator Theory and Operator Algebras

The present article is a review of recent developments concerning the notion of Folner sequences both in operator theory and operator algebras. We also give a new direct proof that any essentially normal operator has an increasing Folner sequence \(\begin{array}{lll}\left\{P_{n}\right\}\end{array}\) of non-zero finite rank projections that strongly converges to 1. The proof is based on Brown–Douglas–Fillmore theory. We use Folner sequences to analyze the class of finite operators introduced by Williams in 1970. In the second part of this article we examine a procedure of approximating any amenable trace on a unital and separable C*-algebra by tracial states \(Tr(.P_{n})/Tr(P_{n})\) corresponding to a Folner sequence and apply this method to improve spectral approximation results due to Arveson and Bedos. The article concludes with the analysis of C*-algebras admitting a non-degenerate representation which has a Folner sequence or, equivalently, an amenable trace. We give an abstract characterization of these algebras in terms of unital completely positive maps and define Folner C*-algebras as those unital separable C*-algebras that satisfy these equivalent conditions. This is analogous to Voiculescu’s abstract characterization of quasidiagonal C*-algebras.

H ∞ -functional calculus and models of Nagy–Foiaş type for sectorial operators

We prove that a sectorial operator admits an H∞-functional calculus if and only if it has a functional model of Nagy–Foiaş type. Furthermore, we give a concrete formula for the characteristic function (in a generalized sense) of such an operator. More generally, this approach applies to any sectorial operator by passing to a different norm (the McIntosh square function norm). We also show that this quadratic norm is close to the original one, in the sense that there is only a logarithmic gap between them.

The Littlewood—Paley—Rubio de Francia property of a Banach space for the case of equal intervals

Let

Resolvent criteria for similarity to a normal operator with spectrum on a curve

X

Infinite-dimensional features of matrices and pseudospectra

be a Banach space. It is proved that an analogue of the Rubio de Francia square function estimate for partial sums of the Fourier series of

Decay rate estimations for linear quadratic optimal regulators

X

Control Of Isomerization In Ensembles Of Nonrigid Molecules based on Classical and Quantum-mechanical Models, LiCN

-valued functions holds true for all disjoint collections of subintervals of the set of integers of equal length and for all exponents

Estimates for nonlinear harmonic ``measures'' on trees

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