On the Commutant Lifting Theorem and Hankel Operators

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.

<p>The classical Nevanlinna-Pick interpolation theorem, proved in 1915 by Pick and in 1919 by Nevanlinna, gives a condition for when there exists an interpolating function in H∞(D) for a specified set of data in the complex plane. In 1967, Sarason proved his commutant lifting theorem for H∞(D), from which an operator theoretic proof of the classical Nevanlinna-Pick theorem followed. Several competing noncommutative generalizations arose as a consequence of Sarason's result, and two strategies emerged for proving generalized Nevanlinna-Pick theorems: via a commutant lifting theorem or via a resolvent, or displacement, equation.</p><p>We explore the difference between these two approaches. Specifically, we compare two theorems: one by Constantinescu-Johnson from 2003 and one by Muhly-Solel from 2004. Muhly-Solel's theorem is stated in the highly general context of W*-correspondences and is proved via commutant lifting. Constantinescu-Johnson's theorem, while stated in a less general context, has the advantage of an elegant proof via a displacement equation. In order to make the comparison, we first generalize Constantinescu-Johnson's theorem to the setting of W*-correspondences in Theorem 3.0.1. Our proof, modeled after Constantinescu-Johnson's, hinges on a modified version of their displacement equation. Then we show that Theorem 3.0.1 is fundamentally different from Muhly-Solel's. More specifically, interpolation in the sense of Muhly-Solel's theorem implies interpolation in the sense of Theorem 3.0.1, but the converse is not true. Nevertheless, we identify a commutativity assumption under which the two theorems yield the same result.</p><p>In addition to the two main theorems, we include smaller results that clarify the connections between the notation, space of interpolating maps, and point evaluation employed by Constantinescu-Johnson and those employed by Muhly-Solel. We conclude with an investigation of the relationship between Theorem 3.0.1 and Popescu's generalized Nevanlinna-Pick theorem proved in 2003.</p>

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.

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.

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

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.

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 introduce a new approach to Nehari’s problem. This approach is based on some kind of fixed point theorem and allows us to obtain some useful generalizations of Nehari’s and Adamyan — Arov — Krein (AAK) theorems. Among those generalizations: descriptions of Hankel operators in weighted ℓ2 spaces; descriptions of Hankel operators from Dirichlet type spaces to weighted Bergman spaces; commutant lifting theorem for non-contractive operators; AAK theorem for “four block operator” etc.KeywordsBergman SpaceSelfadjoint OperatorHankel OperatorKrein SpaceBlock OperatorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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]).

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.

On certain minimal entropy extensions appearing in dilation theory. I