Commutant lifting theorem and interpolation in discrete nest algebras

Ball in [Ba] showed that the commutant lifting theorem for the nest algebras due to Paulsen and Power gives a unified approach to a wide class of interpolation problems for nest algebras. By restricting our attention to the case when nest algebras associated with the problems are discrete we derive a variant of the commutant lifting theorem which avoids language of representation theory and which is sufficient to treat an analog of the generalized Schur-Nevannlinna-Pick (SNP) problem in the setting of upper triangular operators.     

Generalized multipliers for left-invertible analytic operators and their applications to commutant and reflexivity

We introduce generalized?multipliers for?left-invertible analytic operators. We show that they form a Banach algebra and characterize the commutant of such operators in its terms. In the special case, we describe the commutant of balanced weighted shift only in terms of its weights. In addition, we prove two independent criteria for reflexivity of weighted shifts on directed trees.     

Lifting strong commutants of unbounded subnormal operators

Various theorems on lifting strong commutants of unbounded subnormal (as well as formally subnormal) operators are proved. It is shown that the strong symmetric commutant of a closed symmetric operatorS lifts to the strong commutant of some tight selfadjoint extension ofS. Strong symmetric commutants of orthogonal sums of subnormal operators are investigated. Examples of (unbounded) irreducible subnormals, pure subnormals with rich strong symmetric commutants and cyclic subnormals with highly nontrivial strong commutants are discussed.This work was supported by the KBN grant # 2P03A 041 10.     

Relaxation of metric constrained interpolation and a new lifting theorem

In this paper a new lifting interpolation problem is introduced and an explicit solution is given. The result includes the commutant lifting theorem as well as its generalizations in [27] and [2]. The main theorem yields explicit solutions to new natural variants of most of the metric constrained interpolation problems treated in [9]. It is also shown that via an infinite dimensional enlargement of the underlying geometric structure a solution of the new lifting problem can be obtained from the commutant lifting theorem. However, the new setup presented obtained from the commutant lifting theorem. However, the new setup presented in this paper appears to be better suited to deal with interpolations problems from systems and control theory than the commutant lifting theorem.Dedicated to Israel Gohberg, as a token of admiration for his inspiring work in analysis and operator theory, with its far reaching applications, in friendship and with great affection.     

On The Extended Eigenvalues of Some Volterra Operators

We show that a large class of compact quasinilpotent operators has extended eigenvalues. As a consequence, if V is such an operator, then the associated spectral algebra contains its commutant {V}' as a proper subalgebra.     

Bicommutants of Toeplitz operators

In this paper we discuss an unusual phenomenon in the context of Toeplitz operators in the Bergman space on the unit disc: If two Toeplitz operators commute with a quasihomogeneous Toeplitz operator, then they commute with each other. In the Bourbaki terminology, this result can be stated as follows: The commutant of a quasihomogeneous Toeplitz operator is equal to its bicommutant. Received: 11 March 2008     

Characterization Theorems for the Quantum White Noise Gross Laplacian and Applications

This paper reports on the characterization of the quantum white noise (QWN) Gross Laplacian based on nuclear algebra of white noise operators acting on spaces of entire functions with \(\theta \)-exponential growth of minimal type. First, we use extended techniques of rotation invariance operators, the commutation relations with respect to the QWN-derivatives and the QWN-conservation operator. Second, we employ the new concept of QWN-convolution operators. As application, we study and characterize the powers of the QWN-Gross Laplacian. As for their associated Cauchy problem it is solved using a QWN-convolution and Wick calculus.     

Reflexivity of the commutant and local commutants of an algebraic operator

We show that an algebraic operator on a complex Banach space has reflexive commutant if and only if each zero of the minimal polynomial of the operator is simple. Further, for any operator, the local commutant at an eigenvector is reflexive. On the other hand, for an algebraic operator whose minimal polynomial has at least one zero that is not simple, the local commutant of the operator at a given vector is reflexive precisely when the vector is an eigenvector.     

Integral Equations and Operator Theory

A complex number λ is an extended eigenvalue of an operator A if there is a nonzero operator X such that AX = λ XA. We characterize the set of extended eigenvalues, which we call extended point spectrum, for operators acting on finite dimensional spaces, finite rank operators, Jordan blocks, and C₀ contractions. We also describe the relationship between the extended eigenvalues of an operator A and its powers. As an application, we show that the commutant of an operator A coincides with that of Aⁿ, n ≥ 2, n ∈ N if the extended point spectrum of A does not contain any n–th root of unity other than 1. The converse is also true if either A or A^* has trivial kernel.     

Extreme bosonic linear channels

The set of all channels with a fixed input and output is convex. We first give a convenient formulation of the necessary and sufficient condition for a channel to be an extreme point of this set in terms of the complementary channel, a notion of great importance in quantum information theory. This formulation is based on the general approach to extremality of completely positive maps in an operator algebra in the spirit of Arveson. We then use this formulation to prove our main result: under certain nondegeneracy conditions, environmental purity is necessary and sufficient for the extremality of a bosonic linear (quasifree) channel. It hence follows that a Gaussian channel between finite-mode bosonic systems is extreme if and only if it has minimum noise.     

Beyond hyperinvariance for compact operators

We introduce a new class of operator algebras on Hilbert space. To each bounded linear operator a spectral algebra is associated. These algebras are quite substantial, each containing the commutant of the associated operator, frequently as a proper subalgebra. We establish several sufficient conditions for a spectral algebra to have a nontrivial invariant subspace. When the associated operator is compact this leads to a generalization of Lomonosov's theorem.     

K-group and similarity classification of operators

Let H be a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator A in L(H) is said to be a Cowen-Douglas operator if there exist Ω, a connected open subset of complex plane C, and n, a positive integer, such that

(a)

(b)

  for z in Ω;

(c)

; and

(d)

for z in Ω.

In the paper, we give a similarity classification of Cowen-Douglas operators by using the ordered K-group of the commutant algebra as an invariant, and characterize the maximal ideals of the commutant algebras of Cowen-Douglas operators. The theorem greatly generalizes the main result in (Canada J. Math. 156(4) (2004) 742) by simply removing the restriction of strong irreducibility of the operators. The research is also partially inspired by the recent classification theory of simple AH algebras of Elliott-Gong in (Documenta Math. 7 (2002) 255; On the classification of simple inductive limit C^*-algebras, II: The isomorphism theorem, preprint.) (also see (J. Funct. Anal. (1998) 1; Ann. Math. 144 (1996) 497; Amer. J. Math. (1996) 187)).     

Unitary mappings between multiresolution analyses of L2 (R) and a parametrization of low-pass filters

This article provides classes of unitary operators of L²(R) contained in the commutant of the Shift operator, such that for any pair of multiresolution analyses of L²(R) there exists a unitary operator in one of these classes, which maps all the scaling functions of the first multiresolution analysis to scaling functions of the other. We use these unitary operators to provide an interesting class of scaling functions. We show that the Dai-Larson unitary parametrization of orthonormal wavelets is not suitable for the study of scaling functions. These operators give an interesting relation between low-pass filters corresponding to scaling functions, which is implemented by a special class of unitary operators acting on L²([−π, π)), which we characterize. Using this characterization we recapture Daubechies' orthonormal wavelets bypassing the spectral factorization process. Acknowledgements and Notes. Partially supported by NSF Grant DMS-9157512, and Linear Analysis and Probability Workshop, Texas A&M University Dedicated to the memory of Professor Emeritus Vassilis Metaxas.     

Interpolation operators associated with sub-frame sets

Interpolation operators associated with wavelets sets introduced by Dai and Larson play an important role in their operator algebraic approach to wavelet theory. These operators are also related to the von Neumann subalgebras in the ``local commutant' space, which provides the parametrizations of wavelets. It is a particularly interesting question of how to construct operators which are in the local commutant but not in the commutant. Motivated by some questions about interpolation family and C*-algebras in the local commutant, we investigate the interpolation partial isometry operators induced by sub-frame sets. In particular we introduce the -congruence domain of the associated mapping between two sub-frame sets and use it to characterize these partial isometries in the local commutant. As an application, we obtain that if two wavelet sets have the same -congruence domain, then one is a multiresolution analysis (MRA) wavelet set which implies that the other is also an MRA wavelet set.

     

Characterization of the QWN-conservation operator and applications

Based on the finding that the quantum white noise (QWN) conservation operator is a Wick derivation operator acting on white noise operators, we characterize the aforementioned operator by using an extended techniques of rotation invariance operators in a first place. In a second place, we use a new idea of commutation relations with respect to the QWN-derivatives. Eventually, we use the action on the number operator. As applications, we invest these results to study three types of Wick differential equations.     

Triangular strictly cyclic operators

The title refers to an empty class of operators. Moreover, if T is a triangular Banach space operator, then either T is algebraic and the double commutant has infinite strict multiplicity, or T is not algebraic and the commutant has infinite strict multiplicity. A rationally strictly cyclic, but not strictly cyclic, operator cannot have finite strict multiplicity.This research was partially supported by a Grant of the National Science Foundation.     

Unbounded commutants and intertwining spaces of unbounded symmetric operators and 1-representations

This paper investigates the strong commutant, the weak commutant and the form commutant of an unbounded symmetric (nonself-adjoint) operator and of an unbounded 1-representation on a Hilbert space. For two examples of unbounded symmetric operators these commutants are described in terms of singular integral operators and of Toeplitz operators, respectively.     

Extensions of normal operators

We prove that every one dimensional extension of a separably acting normal operator has a cyclic commutant, and that every non-algebraic normal operator has a two-dimensional extension which fails to have a cyclic commutant. Contrasting this, we prove that ifT is an extension of a normal operator by an algebraic operator then the weakly closed algebraW(T) has a separating vector.Partially supported by NSF Grant DMS-9107137