Nevanlinna Pick Type Interpolation Research Articles

Commutant Lifting and Nevanlinna–Pick Interpolation in Several Variables

This paper concerns a commutant lifting theorem and a Nevanlinna–Pick type interpolation result in the setting of multipliers from vector-valued Drury–Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in \({\mathbb {C}}^n\). The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman space and weighted Bergman spaces over the unit ball.

Extremal solutions of matricial Carathéodory problem and Nevanlinna–Pick-type interpolation for Carathéodory matrix functions in the nondegenerate case

This article is devoted to the investigation of remarkable properties of extremal solutions F [U] and G [U] with a q × q unitary parameter U and their distinguished one-parameter families and with |w| = 1 within the solution sets of Problem (C) (the matricial Carathéodory problem) and Problem (NP) (a certain Nevanlinna–Pick-type interpolation for Carathéodory matrix functions), respectively, in the nondegenerate case. We show connections between the poles of the functions F [U] and at the unit circle of the complex plane and their Riesz–Herglotz measures, and discuss closely related matters for the extremal solutions F [U] and . These results further serve as a starting point to look for corresponding properties of the functions G [U] and based on the so-called modified block Toeplitz vector approach.

A multiple point Nevanlinna-Pick type interpolation problem with both interior and boundary data for Carathéodory matrix-valued functions

The main theme of this paper is a study of a multiple point Nevanlinna-Pick type interpolation problem with both interior and boundary data for Caratheodory matrix-valued functions. The so-called Toeplitz vector approach is applied to expose intrinsic connections between such an interpolation problem and a certain truncated trigonometric matrix moment problem with specified constraints that the nonnegative matrix-valued measure has no mass distributions at the boundary interpolation nodes. These connections, together with recent results due to Bolotnikov and Dym, enable us to get solvability criteria for both the Nevanlinna-Pick interpolation problem and the matrix moment problem under consideration. On the basis of the developed theory of matrix moments we can derive parameterized descriptions of all the solutions of these two problems in the nondegenerate case. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Minimum w-entropy interpolants for matricial Carathéodory functions and maximum determinant completions of associated block Pick matrix

The so-called modified block Toeplitz vector approach is used to connect a class of particular solutions G w for w ∈ D of a nondegenerate interpolation problem of the Nevanlinna–Pick type with a class of particular solutions F w of a certain matricial Carathéodory coefficient problem in a transparent way. This will suggest a simple approach to the minimum w-entropy interpolants and the maximum determinant completions of the associated block Pick matrix within the framework of that Nevanlinna–Pick type interpolation problem by using the known assertions corresponding to F w . It turns out that G w ( w ∈ D ) is exactly or provides us with the unique solution to these two extremal problems in a manner.

Functional calculus under Kreiss type conditions

AbstractIt is shown that for an algebraic Banach space operator T , the Kreiss condition, ‖(zI – T )–1‖ ≤ $ {C \over {| z | - 1} } $, |z | > 1, implies the following functional calculus estimate where deg(T ) is the degree of the minimal polynomial annihilating T . This result extends the known estimates of the powers of T for Kreiss operators on finite dimensional spaces. In the case of a general Kreiss operator, an estimate of the rational calculus is proved: Similar estimates hold for the polynomial calculus under generalized Kreiss conditions. A link is also established between the sharp constant in the first estimate and the norm of the best solution for a Nevanlinna–Pick type interpolation problem in analytic Besov classes. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Solution of a multiple Nevanlinna–Pick problem via orthogonal rational functions

We consider a Nevanlinna–Pick type interpolation problem for Carathéodory functions, where the values of the function and its derivatives up to certain orders are given at finitely many points of the unit disk. The set of all solutions of this problem is described by means of the orthogonal rational functions which play here a similar role as the orthogonal polynomials on the unit circle in the classical case of the trigonometric moment problem. In particular, we use a connection between Szegö and Schur parameters which in the classical situation was discovered by Ja.L. Geronimus.

Multivariable Nehari problem and interpolation

In this paper we obtain a multivariable Nehari type theorem for Hankel operators, extending the classical result as well as Treil–Volberg generalization. The result is based on a new generalization of the noncommutative commutant lifting theorem which extends to several variables the commutant lifting result obtained by Treil and Volberg, and recently generalized by Biswas, Foiaş, and Frazho. An extension of I.S. Iokhvidov–Ky Fan theorem is obtained and used to prove a multivariable Adamian–Arov–Krein type theorem for generalized Hankel operators. As consequences, we obtain Carathéodory and Nevanlinna–Pick type interpolation results for “meromorphic” operators on Fock spaces (resp. operator-valued meromorphic functions on the unit ball of C n ). The Nehari type results are used to obtain descriptions of Hankel operators acting on Fock spaces or weighted Fock spaces (multivariable Dirichlet type spaces) and to solve new “weighted” interpolation problems (e.g. Sarason) for noncommutative analytic Toeplitz algebras, which have consequences to the operator-valued analytic interpolation in the unit ball of C n . Moreover, using central intertwining liftings, we obtain explicit formulas for certain solutions of the weighted interpolation problems. As another application of the generalized noncommutative commutant lifting theorem, we derive an abstract interpolation problem for multipliers. This is used to solve a left tangential Nevanlinna–Pick type interpolation problem with operatorial argument for noncommutative analytic Toeplitz algebras.

Meromorphic interpolation in several variables

We obtain a multivariable Nudel’man type interpolation theorem for meromorphic operators on Fock spaces and for operator-valued meromorphic functions on the open unit ball of C n ( n⩾1). As consequences, we provide new Carathéodory–Fejér, Nevanlinna–Pick type interpolation results, and Rosenblum–Rovnyak type restriction theorem for meromorphic operators. In the particular case when n=1, our indefinite generalization of Nudel’man’s theorem provides new interpolation results for meromorphic functions having a finite number of poles in the open unit disc.

Central Intertwining Lifting, Suboptimization, and Interpolation in Several Variables

In this paper we obtain an explicit central intertwining lifting for row contractions. This is used to prove Kaftal–Larson–Weiss and Foiaş–Frazho H∞−H2 suboptimization type results for the noncommutative (resp. commutative) analytic Toeplitz algebra F∞n (resp. W∞n). The algebra F∞n (resp. W∞n) can be viewed as a multivariable noncommutative (resp. commutative) analogue of the Hardy space H∞. Similar results are provided for F∞n⊗B(K, K′) and W∞n⊗B(K, K′), where B(K, K′) is the set of all bounded linear operators acting on Hilbert spaces. New extensions of the Sarason, Carathéodory, and Nevanlinna–Pick type interpolation results are obtained for F∞n⊗B(K, K′) and some consequences to the operator-valued analytic interpolation in the unit ball of Cn are considered.

Nevanlinna-Pick interpolation for non-commutative analytic Toeplitz algebras

The non-commutative analytic Toeplitz algebra is the WOT-closed algebra generated by the left regular representation of the free semigroup onn generators. We obtain a distance formula to an arbitrary WOT-closed right ideal and thereby show that the quotient is completely isometrically isomorphic to the compression of the algebra to the orthogonal complement of the range of the ideal. This is used to obtain Nevanlinna-Pick type interpolation theorems

Multiple point interpolation in Nevanlinna classes

A Nevanlinna-Pick type interpolation problem for generalized Nevanlinna functions is considered. We prescribe the values of the function and its derivatives up to a certain order at finitely many points of the upper half plane. An operator theoretic approach is used to parametrize the solutions of this interpolation problem by means of selfadjoint extensions of a certain symmetry.

Nevanlinna–Pick Type Interpolation in a Dual Algebra

The duality approach of Sarason, Abrahamse, Ball, and others is extended to obtain a verion of Nevanlinna–Pick interpolation for a weak star closed algebra of operators on Hilbert space. In this context factorization is the polar decomposition of a positive trace class operator. Several concrete Nevanlinna–Pick type theorems are seen to follow from the main result and various versions of the Wold decomposition.

Nehari and Nevanlinna-Pick Problems and Holomorphic Extensions in the Polydisk in Terms of Restricted BMO

A new class BMOr is introduced, motivated by the following extension of the Nehari theorem to d ≥ 1: The norm of a bounded (big) Hankel form B in the d-dimensional torus satisfies || φ|| ≤ || B|| ≤ √ d || φ || where the norm of φ, the antianalytic symbol of B, is taken in BMOr. This extension in turn provides solutions to Nevanlinna-Pick type interpolation problems in the polydisk, with control of the BMOr norm. Necessary and sufficient conditions for the existence of H 2 ∩ BMOr extensions of analytic functions defined on a zero variety V are given, with control of the norm, under general conditions on V.