Stephen C. Power

Schur Products and Matrix Completions

We prove that a necessary and sufficient condition for a given partially positive matrix to have a positive completion is that a certain Schur product map defined on a certain subspace of matrices is a positive map. By analyzing the positive elements of this subspace, we obtain new proofs of the results of H. Dym and I. Gohberg and Grone, Johnson, Sa, and Wolkowitz (Linear Algebra Appl.58 (1984), 109–124). (Linear Algebra Appl.36 (1981), 1–24). We also obtain a new proof of a result of U. Haagerup (Decomposition of completely bounded maps on operation algebras, preprint), characterizing the norm of Schur product maps, and a new Hahn-Banach type extension theorem for these maps. Finally, we obtain generalizations of many of these results to matrices of operators, which we apply to the study of representations of certain subalgebras of the n × n matrices.

Factorization in analytic operator algebras

Abstract A constructive and unified approach is used to obtain the upper-lower factorization of positive operators and the outer function factorization of positive operator valued functions on the circle. For a projection nest E it is shown that every positive operator admits a canonical factorization C = A ∗ A , with A an outer operator, if and only if E is well ordered. With new methods we generalize the inner-outer factorizations obtained by Arveson, for nests of order type Z , and the Riesz factorization, due to Shields, for trace class triangular operators. Weak factorization is obtained in noncommutative H1 spaces associated with (general) nest subalgebras of a semifinite factor. Characterizations of a Nehari type are given for the associated Hankel forms and Hankel operators.

C∗-algebras generated by Hankel operators and Toeplitz operators

Abstract Results relating the spectra and essential spectra of Hankel operators to their symbols are obtained by various considerations of the C∗-algebras that they generate. Such considerations exploit the elementary relationships between Hankel operators and Toeplitz operators and established techniques in the theory of Toeplitz operators and their generated C∗-algebras.

Operators and function theory

Bloch Functions: The Basic Theory.- A Survey of Some Results on Subnormal Operators.- Optimization, Engineering, and a More General Corona Theorem.- Minimal Factorization, Linear Systems and Integral Operators.- Ha-Plitz Operators: A Survey of Some Recent Results.- Stochastic Processes, Infinitesimal Generators and Function Theory.- Paracommutators and Minimal Spaces.- Decomposition Theorems for Bergman Spaces and their Applications.- Operator-Theoretic Aspects of the Nevanlinna-Pick Interpolation Problem.- Cyclic Vectors in Banach Spaces of Analytic Functions.- Interpolation by Analytic Matrix Functions.

Subalgebras of Graph C*-algebras

I give a self-contained introduction to two novel classes of nonselfadjoint operator algebras, namely the generalised analytic Toeplitz algebras L G , associated with the “Fock space” of a graph G, and subalgebras of graph C*-algebras. These two topics are somewhat independent but in both cases I shall focus on fundamental techniques and problems related to classifying isomorphism types and to the recovery of underlying foundational structures, be they graphs or groupoids.

Finite rank multivariable Hankel forms

Abstract We obtain a several variables generalization of Kroneckers well-known result on finite rank Hankel matrices.

The Fourier binest algebra.

The Fourier binest algebra is dened as the intersection of the Volterra nest al- gebra on L 2 (R) with its conjugate by the Fourier transform. Despite the absence of nonzero nite rank operators this algebra is equal to the closure in the weak oper- ator topology of the Hilbert{Schmidt bianalytic pseudo-differential operators. The (non-distributive) invariant subspace lattice is determined as an augmentation of the Volterra and analytic nests (the Fourier binest) by a continuum of nests associated with the unimodular functions exp( isx 2 =2) for s> 0. This multinest is the reflex- ive closure of the Fourier binest and, as a topological space with the weak operator topology, it is shown to be homeomorphic to the unit disc. Using this identication the unitary automorphism group of the algebra is determined as the semi-direct productR 2 R for the action t(; ) =( e t ;e t ):

The failure of approximate inner conjugacy for standard diagonals in regular limit algebras.

AF C*-algebras contain natural AF masas which, here, we call standard diagonals. Standard diagonals are unique, in the sense that two standard diagonals in an AF C*-algebra are conjugate by an approximately inner automorphism. We show that this uniqueness fails for non-selfadjoint AF operator algebras. Precisely, we construct two standard diagonals in a particular non-selfadjoint AF operator algebra which are not conjugate by an approximately inner automorphism of the non-selfadjoint algebra.

Semi-discreteness and dilation theory for nest algebras

Abstract In this paper it is shown that a contractive σ-weakly continuous Hilbert space representation of a nest algebra admits a σ-weakly continuous dilation to the containing algebra of all operators. This is accomplished by first establishing the complete contractivity of contractive representations through a semi-discreteness property for nest algebras relative to finite-dimensional nest algebras (Theorem 2.1). The semi-discreteness property is obtained by an examination of the order type, spectral type, and multiplicity of the nest, and by the construction of subalgebras that are completely isometric copies of finite-dimensional nest algebras, with good approximation properties. With complete contractivity at hand, the desired dilation follows from Arvesons dilation theorem and auxiliary arguments.

Completely contractive representations for some doubly generated antisymmetric operator algebras

Contractive weak star continuous representations of the Fourier binest algebra (of Katavolos and Power) are shown to be completely contractive. The proof depends on the approximation of by semicrossed product algebras and on the complete contractivity of contractive representations of such algebras. The latter result is obtained by two applications of the Sz.-Nagy-Foias lifting theorem. In the presence of an approximate identity of compact operators it is shown that an automorphism of a general weakly closed operator algebra is necessarily continuous for the weak star topology and leaves invariant the subalgebra of compact operators. This fact and the main result are used to show that isometric automorphisms of the Fourier binest algebra are unitarily implemented.