Tavan T. Trent

Interpolation and Commutant Lifting for Multipliers on Reproducing Kernel Hilbert Spaces

We obtain an explicit representation formula and a Nevanlinna-Pick-type interpolation theorem for the multiplier space of the reproducing kernel space ℌ(k d ) of analytic functions on the d-dimensional complex unit ball with reproducing kernel k d (z, w) = 1/(1 — (z, w)). More generally, if k is a positive kernel on a set Ω such that 1/k has 1 positive square, then there is an embedding b of Ω into the ball B d (where d is the number of negative squares of 1/k) such that any multiplier W for ℌ (k) lifts to a multiplier F for the space ℌ (kd) on the ball (W = F o b). As a corollary, multipliers for ℌ (k) also have explicit realization formulas and satisfy a Nevanlinna-Pick-type interpolation theorem. All the results in fact extend to the case of matrix-and operator-valued multipliers and left tangential Nevanlinna-Pick interpolation. Contractive multiplier solutions of a given set of interpolation conditions correspond to unitary extensions of a partially defined isometric operator; hence a technique of Arov and Grossman can be used to give a linear-fractional parametrization for the set of all such interpolants. A more abstract formulation of the analysis leads to a commutant lifting theorem for such multipliers. In particular, we obtain a new proof of a result of Quiggin giving sufficient conditions on a kernel k for a Nevanlinna-Pick-type interpolation theorem to hold for the space of multipliers on a reproducing kernel Hilbert space ℌ (k).

Toeplitz Corona Theorems for the Polydisk and the Unit Ball

Abstract.The main purpose of this paper is to extend and refine some work of Agler–McCarthy and Amar concerning the Corona problem for the polydisk and the unit ball in

The Abstract Interpolation Problem and Commutant Lifting: A Coordinate-free Approach

\mathbb{C}^n

Solutions for the H∞(Dn) Corona Problem Belonging to exp(L1/2n-1

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Operator Theory and the Corona Problem on the Bidisk

We show that a usual corona-type theorem on a space of functions automatically extends to a matrix version.

An H2-corona theorem on the bidisk for infinitely many functions☆

We present a coordinate-free formulation of the Abstract Interpolation Problem introduced by Katsnelson, Kheifets and Yuditskii in an abstract scattering theory framework. We also show how the commutant lifting theorem fits into this new formulation of the Abstract Interpolation Problem, giving a coordinate-free version of a result of Kupin.

Factoring positive operators on reproducing kernel Hilbert spaces

Abstract We consider linear surjective isometries acting on reflexive operator algebras with commutative, completely distributive subspace lattices. Such isometries come in two types: those implemented by unitary operators and conjugations acting on the Hilbert space, and those implemented by unitaries alone. It is shown that every isometry is a direct sum of these types.

A separating problem on function spaces

For a countable number of input functions in H ∞ (D n ), we find explicit analytic solutions belonging to the Orlicz-type space, \( \exp (L^{\tfrac{1} {{2n - 1}}} ) \). Note that \( H^\infty (D^n ) - BMO(D^n ) \subsetneqq \exp (L^{\tfrac{1} {{2n - 1}}} ) \subsetneqq \cap _1^\infty H^p (D^n ) \)