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A commutant lifting theorem on analytic polyhedra

Calin Ambrozie, Jörg Eschmeier (2005)

Banach Center Publications

In this note a commutant lifting theorem for vector-valued functional Hilbert spaces over generalized analytic polyhedra in ℂⁿ is proved. Let T be the compression of the multiplication tuple M z to a *-invariant closed subspace of the underlying functional Hilbert space. Our main result characterizes those operators in the commutant of T which possess a lifting to a multiplier with Schur class symbol. As an application we obtain interpolation results of Nevanlinna-Pick and Carathéodory-Fejér type...

An operator-theoretic approach to truncated moment problems

Raúl Curto (1997)

Banach Center Publications

We survey recent developments in operator theory and moment problems, beginning with the study of quadratic hyponormality for unilateral weighted shifts, its connections with truncated Hamburger, Stieltjes, Hausdorff and Toeplitz moment problems, and the subsequent proof that polynomially hyponormal operators need not be subnormal. We present a general elementary approach to truncated moment problems in one or several real or complex variables, based on matrix positivity and extension. Together...

Bounds for Fractional Powers of Operators in a Hilbert Space and Constants in Moment Inequalities

I. Gil’, Michael (2009)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 47A56, 47A57,47A63We derive bounds for the norms of the fractional powers of operators with compact Hermitian components, and operators having compact inverses in a separable Hilbert space. Moreover, for these operators, as well as for dissipative operators, the constants in the moment inequalities are established.* This research was supported by the Kamea Fund of Israel.

Fejér–Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle

Jeffrey S. Geronimo, Plamen Iliev (2014)

Journal of the European Mathematical Society

We give a complete characterization of the positive trigonometric polynomials Q ( θ , ϕ ) on the bi-circle, which can be factored as Q ( θ , ϕ ) = | p ( e i θ , e i ϕ ) | 2 where p ( z , w ) is a polynomial nonzero for | z | = 1 and | w | ≤ 1 . The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight 1 4 π 2 Q ( θ , ϕ ) on the bi-circle. We use this result to describe how specific factorizations of weights on the bi-circle can be translated into identities relating...

Generalized Fock spaces, interpolation, multipliers, circle geometry.

Jaak Peetre, Sundaram Thangavelu, Nils-Olof Wallin (1996)

Revista Matemática Iberoamericana

By a (generalized) Fock space we understand a Hilbert space of entire analytic functions in the complex plane C which are square integrable with respect to a weight of the type e-Q(z), where Q(z) is a quadratic form such that tr Q > 0. Each such space is in a natural way associated with an (oriented) circle C in C. We consider the problem of interpolation between two Fock spaces. If C0 and C1 are the corresponding circles, one is led to consider the pencil of circles generated by C0 and C1....

Interpolation by elementary operators

Bojan Magajna (1993)

Studia Mathematica

Given two n-tuples a = ( a 1 , . . . , a n ) and b = ( b 1 , . . . , b n ) of bounded linear operators on a Hilbert space the question of when there exists an elementary operator E such that E a j = b j for all j =1,...,n, is studied. The analogous question for left multiplications (instead of elementary operators) is answered in any C*-algebra A, as a consequence of the characterization of closed left A-submodules in A n .

L¹ factorizations, moment problems and invariant subspaces

Isabelle Chalendar, Jonathan R. Partington, Rachael C. Smith (2005)

Studia Mathematica

For an absolutely continuous contraction T on a Hilbert space 𝓗, it is shown that the factorization of various classes of L¹ functions f by vectors x and y in 𝓗, in the sense that ⟨Tⁿx,y⟩ = f̂(-n) for n ≥ 0, implies the existence of invariant subspaces for T, or in some cases for rational functions of T. One of the main tools employed is the operator-valued Poisson kernel. Finally, a link is established between L¹ factorizations and the moment sequences studied in the Atzmon-Godefroy method, from...

On a Five-Diagonal Jacobi Matrices and Orthogonal Polynomials on Rays in the Complex Plane

Zagorodniuk, S. (1998)

Serdica Mathematical Journal

∗ Partially supported by Grant MM-428/94 of MESC.Systems of orthogonal polynomials on the real line play an important role in the theory of special functions [1]. They find applications in numerous problems of mathematical physics and classical analysis. It is known, that classical polynomials have a number of properties, which uniquely define them.

On a general bidimensional extrapolation problem

Rodrigo Arocena, Fernando Montana (1993)

Colloquium Mathematicae

Several generalized moment problems in two dimensions are particular cases of the general problem of giving conditions that ensure that two isometries, with domains and ranges contained in the same Hilbert space, have commutative unitary extensions to a space that contains the given one. Some results concerning this problem are presented and applied to the extension of functions of positive type.

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