Pick interpolation, displacement equations, and W*-correspondences

The commutant lifting theorem of [24] may be regarded as a very general interpolation theorem from which a number of classical interpolation results may be deduced. In this paper we prove a spectral version of the commutant lifting theorem in which one bounds the spectral radius of the interpolant and not the norm. We relate this to a spectral analogue of classical matricial Nevanlinna-Pick interpolation.

This chapter contains the proofs of the operator-valued interpolation theorems introduced in the previous chapter. The proofs are based on the commutant lifting theorem which is stated for convenience in the first section. In Chapter IV the commutant lifting theorem will be revisited in more detail, and a constructive proof of this theorem will be given there.

In this note, we prove a harmonic-type maximal principle for the Schur parametrization of all intertwining liftings of an intertwining contraction in the commutant lifting theorem.

Part 1 Interpolation and time-invariant system: interpolation problems for time-valued functions proofs using the commutant lifting theorem time invariant systems central commutant lifting central state space solutions parametization of intertwinning and its applications applications to control systems. Part 2 Nonstationary interpolation and time-varying systems nonstationary interpolation theorems nonstationary systems and point evaluation reduction techniques - from nonstationary to stationary and vice versa proofs of the nonstationary interpolation theorems by reduction to the stationary case a general completion theorem applications of the three chains completion theorem to interpolation parameterization of all solutions of the three chains completion problem.

In this note we give another proof of the commutant lifting theorem (see) [6] and [7]), based on the Adamjan-Arov-Krein techniques introduced in [1]. We then apply the construction given in Theorem 1 below to obtain a generalization of a result in [2] (see also [4]).

Nonlinearity inH ?-control theory, causality in the commutant lifting theorem, and extension of intertwining operators

Relative Yoneda Cohomology for Operator Spaces

The symmetrized bidisc has been a rich field of holomorphic function theory and operator theory. A certain well-known reproducing kernel Hilbert space of holomorphic functions on the symmetrized bidisc resembles the Hardy space of the unit disc in several aspects. This space is known as the Hardy space of the symmetrized bidisc. We introduce the study of those operators on the Hardy space of the symmetrized bidisc that are analogous to Toeplitz operators on the Hardy space of the unit disc. More explicitly, we first study multiplication operators on a bigger space (an $L^2$-space) and then study compressions of these multiplication operators to the Hardy space of the symmetrized bidisc and prove the following major results. (1) Theorem I analyzes the Hardy space of the symmetrized bidisc, not just as a Hilbert space, but as a Hilbert module over the polynomial ring and finds three isomorphic copies of it as $\mathbb D^2$-contractive Hilbert modules. (2) Theorem II provides an algebraic, Brown and Halmos-type characterization of Toeplitz operators. (3) Theorem III gives several characterizations of an analytic Toeplitz operator. (4) Theorem IV characterizes asymptotic Toeplitz operators. (5) Theorem V is a commutant lifting theorem. (6) Theorem VI yields an algebraic characterization of dual Toeplitz operators. Every section from Section 2 to Section 7 contains a theorem each, the main result of that section.

In this note a commutant lifting theorem for vector-valued functional Hilbert spaces over generalized analytic polyhedra in $\Bbb{C}^n$ is proved. Let $T$ be the compression of the multiplication tuple $M_z$ to a $*$-invariant closed subspace of the under

Bergman spaces over noncommutative domains and commutant lifting

The commutant lifting theorem of Sz.-Nagy and Foias ([19]), a cornerstone result in the theory of contractive operators on Hilbert space, has found a large variety of applications in most distinct areas of pure and applied analysis (see [13] and the references within) The existence theorem is complemented by different descriptions of the class of all commutant liftings of a given triple. Subsequently, a major direction of studies have concentrated on describing the properties of distinguished liftings; as interesting cases, let us note the central lifting([14], [8]), or the liftings which correspond in engineering applications to rational realizations([18]).

Proofs of commutativity theorems for general rings usually employ the Jacobson structure theory; however, alternative approaches to the "xn = x theorem" [ l, 2] suggest that the power of the Jacobson theory is not required. In this note we prove two commutativity theorems of Herstein in an elementary way. Both proofs involve establishing first that the rings under consideration are duo-rings - rings in which every one-sided ideal is two-sided.

We develop a dilation theory on noncommutative varieties determined by row contractions T: = [T 1 ,..., T n ] subject to constraints such as p(T 1 ,...,T n ) = 0, p ∈ P, where T is a set of noncommutative polynomials. The model n-tuple is the universal row contraction [B 1 ,...,B n ] satisfying the same constraints as T, which turns out to be, in a certain sense, the maximal constrained piece of the n-tuple [S 1 ,..., S n ] of left creation operators on the full Fock space on n generators. The theory is based on a class of constrained Poisson kernels associated with T and representations of the C*-algebra generated by B 1 ,...,B n and the identity. Under natural conditions on the constraints we have uniqueness for the minimal dilation. A characteristic function Θ T is associated with any (constrained) row contraction T and it is proved that I - Θ T Θ* T = K T K* T , where K T is the (constrained) Poisson kernel of T. Consequently, for pure constrained row contractions, we show that the characteristic function is a complete unitary invariant and provide a model. We show that the curvature invariant and Euler characteristic asssociated with a Hilbert module generated by an arbitrary (resp. commuting) row contraction T can be expressed only in terms of the (resp. constrained) characteristic function of T. We provide a commutant lifting theorem for pure constrained row contractions and obtain a Nevanlinna-Pick interpolation result in our setting.

The central solution for the commutant lifting theorem is used to solve mixed H2-H∞ tangential interpolation problems. These include an H2-H∞ tangential Caratheodory interpolation problem, an H2-H∞ tangential Nevanlinna-Pick interpolation problem and an H2-H∞ tangential Hermite-Fejer interpolation problem. Explicit computational formulas in terms of state space realizations are given.

In this note we examine certain sharpenings of the Tietze-Urysohn extension theorem. We prove, for example, that if A is a closed subspace of a normal space X then there is a continuous extender v,:C*(A) -C*(X), where C*(A) and C*(X) are the Banach spaces of continuous, bounded, real-valued functions on A and X, respectively. However, well-known examples show that we cannot arrange that q is both continuous and linear, or that q is an isometry. Introduction. The classical Tietze-Urysohn theorem guarantees that every continuous, bounded, real-valued function f defined on a closed subspace A of a normal space X can be extended to a continuous, real-valued function f defined on all of X and having the same bounds as f. If we let C*(A) and C*(X) denote the real vector spaces of continuous, bounded, real-valued functions on A and X respectively, then the Axiom of Choice yields the following slight reformulation of the classical theorem. A. THEOREM (Tietze-Urysohn). If A is a closed subspace of the normal space X then there is a function r: C*(A ) C*(X) such that for every f E C*(A), rl(f) extends f and has the same bounds as f Any function q:C*(A)-+ C*(X) having the property that 71(f) extends f whenever f E C*(A) will be called an extender from C*(A) to C*(X). It is natural to wonder whether, if one selects the extension 71(f) of f more carefully, one can obtain an extender r1 which respects the natural linear and/or topological structures of the function spaces in Theorem A. For metric spaces, an affirmative answer is provided by the Dugundji Extension Theorem. B. THEOREM [D]. Suppose A is a closed subset of a metrizable space X. Then there is a function 77: C*(A) -* C*(X) such that: (1) for each f E C*(A), r(f) extends f and has the same bounds as f; (2) the function 77 is linear.(2' Theorem B is a model for all of the extension theorems considered in this paper so we shall pause to make a few comments about its conclusions and its proof. In introducing the theorem we said that it shows that 7q can be constructed in such a way that 7q respects the natural topological structures of the function spaces. Let us now introduce one such structure, the sup-norm topology. For a function f E C*(A) the sup-norm of f is defined by lIf || = sup {if(x){:x E A}; the norm of a function F E C*(X) is analogously defined. When equipped with these norms, C*(A) and C*(X) are Banach spaces and the two conclusions of Theorem B force 77 to be an isometry. That may be seen as follows. Let f, g E C*(A). Then 1j 71(f) 71(g) jj = 1177 (f g) I since 77 is linear, and because 7q(f g) has the same bounds as f g, 1j 71 (f g) jj = jj f g 1j. Thus 7q is a linear, isometric extender. The proof of Theorem B is too technical to include here but its central idea is easily described, at least in principle. One does not extend the functions in C*(A) one at a time as in the proof of Theorem A; instead one defines an extension process which acts simultaneously (and linearly) on all members of C*(A). Dugundji's extension process has been improved in recent years so that it now 1. This paper was completed while the third author was visiting the University of Pittsburgh as a Mellon Postdoctoral Fellow. 2. Dugundji actually proved that a linear q can be found such that for each f E C*(A), v(f) extends f and the range of q (f) is contained in the convex hull of the range of f. Furthermore, if one considers bounded, continuous, complex-valued functions, then one can obtain complex-linear extenders with the property that the range of v1(f) is contained in the convex hull of the range of f.