Douglas Farenick

The spectral theorem in quaternions

An exposition of the spectral theory of normal matrices with quaternion entries is presented.

HYPONORMALITY AND SPECTRA OF TOEPLITZ OPERATORS

This paper concerns algebraic and spectral properties of Toeplitz operators Tφ, on the Hardy space H2(T), under certain assumptions concerning the symbols φ ∈ L∞(T). Among our algebraic results is a characterisation of normal Toeplitz opertors with polynomial symbols, and a characterisation of hyponormal Toeplitz operators with polynomial symbols of a prescribed form. The results on the spectrum are as follows. It is shown that by restricting the spectrum, a set-valued function, to the set of all Toeplitz operators, the spectrum is continuous at Tφ, for each quasicontinuous φ. Secondly, we examine under what conditions a classic theorem of H. Weyl, which has extensions to hyponormal and Toeplitz operators, holds for all analytic functions of a single Toeplitz operator with continuous symbol.

Normal Toeplitz Matrices

It is well known from the work of Brown and Halmos [J. Reine Angew. Math., 213 (1963/1964), pp. 89--102] that an infinite Toeplitz matrix is normal if and only if it is a rotation and translation of a Hermitian Toeplitz matrix. In the present article we prove that all finite normal Toeplitz matrices are either generalised circulants or are obtained from Hermitian Toeplitz matrices by rotation and translation.

Irreducible positive linear maps on operator algebras

Motivated by the classical results of G. Frobenius and O. Perron on the spectral theory of square matrices with nonnegative real entries, D. Evans and R. Høegh-Krohn have studied the spectra of positive linear maps on general (noncommutative) matrix algebras. The notion of irreducibility for positive maps is required for the Frobenius theory of positive maps. In the present article, irreducible positive linear maps on von Neumann algebras are explicitly constructed, and a criterion for the irreducibility of decomposable positive maps on full matrix algebras is given. Let A and A denote a C∗-algebra and its cone of positive elements. A linear map φ : A→ A is positive if it leaves the cone A invariant; that is, φ(x) ∈ A for every x ∈ A. If A is an n-dimensional commutative C∗-algebra, then a positive map φ on A has a representation as an n× n matrix with nonnegative real entries. Therefore, the classical matrix theoretic results (see [7, Ch.8]) of O. Perron and G. Frobenius on matrices with nonnegative entries can be viewed as an important case of a more general theory that deals with positive maps on operator algebras. This is the viewpoint taken by D. Evans and R. Høegh-Krohn [6], S. Albeverio and R. Høegh-Krohn [2], and U. Groh [8], [9] in their works on the spectra of positive maps on operator algebras. In both the classical theory and its various generalisations [12], and in the one put forward in [6], [2], [8], and [9], a positive map that is “strictly” positive, or that is “irreducible”, will possess certain interesting spectral properties. Although the spectral theory of such maps has been studied, the issue of how one is to determine whether a given positive linear map on an operator algebra is strictly positive or irreducible (or neither) has received much less attention. For matrices with nonnegative entries a simple and readily verifiable criterion exists, dating back to Frobenius: φ is strictly positive if and only if each entry of φ is positive, and φ is irreducible if and only if the directed graph of φ is strongly connected. However for positive maps on noncommutative operator algebras, it is somewhat more difficult to make this determination. Indeed this difficulty appears to be hampered even further by the fact that there is no tractable structure theory for positive maps. Received by the editors May 2, 1995. 1991 Mathematics Subject Classification. Primary 46L05.

On hyponormal Toeplitz operators with polynomial and circulant-type symbols

This paper characterises those hyponormal Toeplitz operators on the Hardy space of the unit circle among all Toeplitz operators that have polynomial symbols with circulant-type sets of coefficients.

*-extreme points of some compact *-convex sets

In the C*-algebra Mn of complex n x n matrices, we consider the notion of noncommutative convexity called C*-convexity and the corresponding notion of a C*-extreme point. We prove that each irreducible element of Mn is a C*-extreme point of the C*-convex set it generates, and we classify the C*-extreme points of any C*-convex set generated by a compact set of normal matrices.

The structure of *-extreme points in spaces of completely positive linear maps on *-algebras

If A is a unital C*-algebra and if H is a complex Hilbert space, then the set SH(A) of all unital completely positive linear maps from A to the algebra B(H) of continuous linear operators on H is an operator-valued, or generalised, state space of A. The usual state space of A occurs with the one-dimensional Hilbert space C. The structure of the extreme points of generalised state spaces was determined several years ago by Arveson [Acta Math. 123(1969), 141-224]. Recently, Farenick and Morenz [Trans. Amer. Math. Soc. 349(1997), 1725-1748] studied generalised state spaces from the perspective of noncommutative convexity, and they obtained a number of results on the structure of C*-extreme points. This work is continued in the present paper, and the main result is a precise description of the structure of the C*extreme points of the generalised state spaces of A for all finite-dimensional Hilbert spaces H.

Young’s Inequality in Compact Operators

This paper extends to compact operators an interesting inequality first proved by T. ANDO for matrices: for any compact operators a and b acting on a complex separable Hilbert space, there is a partial isometry u such that \(u\left| {a{b^*}} \right|{u^*}\frac{1}{p}{\left| a \right|^q},\)for every \(p,q \in \left( {1,\infty } \right)\) that satisfy \(1/p + 1/q = 1\)

Classical and nonclassical randomness in quantum measurements

The space POVM H(X) of positive operator-valued probability measures on the Borel sets of a compact (or even locally compact) Hausdorff space X with values in B(H), the algebra of linear operators acting on a d-dimensional Hilbert space H, is studied from the perspectives of classical and nonclassical convexity through a transform Γ that associates any positive operator-valued measure ν with a certain completely positive linear map Γ(ν) of the homogeneous C*-algebra C(X)⊗B(H) into B(H). This association is achieved by using an operator-valued integral in which nonclassical random variables (that is, operator-valued functions) are integrated with respect to positive operator-valued measures and which has the feature that the integral of a random quantum effect is itself a quantum effect. A left inverse Ω for Γ yields an integral representation, along the lines of the classical Riesz representation theorem for linear functionals on C(X), of certain (but not all) unital completely positive linear maps φ:C(X)...

Conditional expectation and Bayes’ rule for quantum random variables and positive operator valued measures

A quantum probability measure ν is a function on a σ-algebra of subsets of a (locally compact and Hausdorff) sample space that satisfies the formal requirements for a measure, but where the values of ν are positive operators acting on a complex Hilbert space, and a quantum random variable is a measurable operator valued function. Although quantum probability measures and random variables are used extensively in quantum mechanics, some of the fundamental probabilistic features of these structures remain to be determined. In this paper, we take a step toward a better mathematical understanding of quantum random variables and quantum probability measures by introducing a quantum analogue for the expected value Eν[ψ] of a quantum random variable ψ relative to a quantum probability measure ν. In so doing we are led to theorems for a change of quantum measure and a change of quantum variables. We also introduce a quantum conditional expectation which results in quantum versions of some standard identities for R...