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.     

On unbounded Bergman operators

This paper explores properties of the Bergman operator on unbounded open subsets of the plane. In addition to the characterization of the bounded commutant of such operators it proves the Berger-Shaw theorem and gives some general criteria under which the operator and its self-commutator are densely defined.     

Sufficiency and strong commutants in quantum probability theory

A probability algebra (A, *, ω) consisting of a^*algebraA with a faithful state ω provides a framework for an unbounded noncommutative probability theory. A characterization of symmetric probability algebra is obtained in terms of an unbounded strong commutant of the left regular representation ofA. Existence of coarse-graining is established for states that are absolutely continuous or continuous in the induced topology. Sufficiency of a^*subalgebra relative to a family of states is discussed in terms of noncommutative Radon-Nikodym derivatives (a form of Halmos-Savage theorem), and is applied to couple of examples (including the canonical algebra of one degree of freedom for Heisenberg commutation relation) to obtain unbounded analogues of sufficiency results known in probability theory over a von Neumann algebra.     

A note on the functional calculus for unbounded self-adjoint operators

Two theorems of Riesz and Lorch (1936) are used to pass directly from the functional calculus for bounded symmetric operators to that for unbounded self-adjoint operators, thereby considerably shortening the passage via the spectral resolution for unbounded self-adjoint operators, and making particularly transparent the manner in which properties of the functional calculus for bounded operators are inherited by those which are unbounded.     

A lifting theorem for unbounded quasinormal operators

Relationships between minimal normal extensions of spectral and cyclic type of an unbounded quasinormal operator are discussed and some properties such as, for example, tightness of such extensions are established. A Yoshino type criterion on the lifting of the strong commutant of an unbounded quasinormal operator is proved.     

Symmetric wiener-hopf and Toeplitz operators

We explain the relationship between the principal function of A C^* algebra generated by a pair of unitary operators with commutator of one dimensional range and the deficiency spaces of symmetric Toeplitz operators defined by real unbounded symbols.     

Some Selfadjoint 2$\times$2 Operator Matrices Associated with Closed Operators

We study a 22 operator matrix associated with a closed densely defined operator. Among others, the selfadjointness of a closed symmetric operator and the strong commutativity of two (unbounded) self-adjoint operators are characterized in terms of the related operator matrices. We propose a definition of strong commutativity for closed symmetric operators. Submitted: November 8, 2001     

完备赋准范空间上一类无界线性随机算子的谱分解定理

本文利用随机内积空间方法给出了完备赋准范空间上一类无界线性随机算子的谱分解定理。此结果不仅推广了对称随机线性算子的谱分解定理，而且限于原情形也使其处理简明、清晰。     

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.     

Solutions périodiques des équations d'évolution

We study here the necessary and sufficient condition for the existence of periodic solutions for evolution equations in the case of linear unbounded maximal monotonous and symmetric operators. We present also an estimate of the convergence time to the periodic states in the finite-dimensional case.     

Commutants modulo the compact operators of certain CSL algebras II

The commutant modulo compacts, or essential commutant, of a reflexive algebra with commutative subspace lattice is a C^* algebra which is the sum of the compact operators in L(H) and a C^* subalgebra of the core. We give a characterization of the essential commutant of a separably acting CSL algebra in terms of properties of the spectral measure of an operator in the intersection of the essential commutant and the core. This is used to determine some sufficient conditions on the lattice for when the essential commutant is norm generated by the projections it contains.     

Operator integrals and sesquilinear forms

We consider various systematic ways of defining unbounded operator valued integrals of complex functions with respect to (mostly) positive operator measures and positive sesquilinear form measures, and investigate their relationships to each other in view of the extension theory of symmetric operators. We demonstrate the associated mathematical subtleties with a physically relevant example involving moment operators of the momentum observable of a particle confined to move on a bounded interval.     

Composition operators: Hyperinvariant subspaces, quasi-normals and isometries

We exhibit hyperinvariant subspaces of some composition operators. We also consider quasi-normal composition operators and discuss the commutant of isometric composition operators.

     

Noiseless Subsystems for Collective Rotation Channels in Quantum Information Theory

Collective rotation channels are a fundamental class of channels in quantum computing and quantum information theory. The commutant of the noise operators for such a channel is a C*-algebra which is equal to the set of fixed points for the channel. Finding the precise spatial structure of the commutant algebra for a set of noise operators associated with a channel is a core problem in quantum error prevention. We draw on methods of operator algebras, quantum mechanics and combinatorics to explicitly determine the structure of the commutant for the class of collective rotation channels.     

On tuples of commuting symmetric,non-selfadjoint operators

Generalizing the Cowen-Douglas-Theory to certain tuples of unbounded symmetric operators we obtain canonical models for such tuples, which are realized in holomorphic functional Hilbert spaces. The results are applied to multidimensional moment problems.     

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.

     

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.     

Indices, characteristic numbers and essential commutants of Toeplitz operators

For an essentially normal operatorT, it is shown that there exists a unilateral shift of multiplicitym inC ^* (T) if and only if γ(T)≠0 and γ(T)/m. As application, we prove that the essential commutant of a unilateral shift and that of a bilateral shift are not isomorphic asC ^* -algebras. Finally, we construct a naturalC ^* -algebra ε + ε_* on the Bergman spaceL _a ² (B _n ), and show that its essential commutant is generated by Toeplitz operators with symmetric continuous symbols and all compact operators. Supported by NSFC and Laboratory of Mathematics for Nonlinear Science at Fudan University.     

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.     

Rank one perturbations of not semibounded operators

Rank one perturbations of selfadjoint operators which are not necessarily semibounded are studied in the present paper. It is proven that such perturbations are uniquely defined, if they are bounded in the sense of forms. We also show that form unbounded rank one perturbations can be uniquely defined if the original operator and the perturbation are homogeneous with respect to a certain one parameter semigroup. The perturbed operator is defined using the extension theory for symmetric operators. The resolvent of the perturbed operator is calculated using Krein's formula. It is proven that every rank one perturbation can be approximated in the operator norm. We prove that some form unbounded perturbations can be approximated in the strong resolvent sense without renormalization of the coupling constant only if the original operator is not semibounded. The present approach is applied to study first derivative and Dirac operators with point interaction, in one dimension.