On Joint Hyponormality of Operators

Abstract

ABSTRACT. A notion of joint hyponormality is introduced for a collection of bounded linear operators on a separable Hubert space.It is the purpose of this note to introduce a notion of joint hyponormality for a collection of bounded linear operators on a separable Hubert space ßtf in a way that will meet the following conditions.(a) The notion of joint hyponormality will be in some sense a natural generalization of the notion of hyponormality for a single operator.(b) The notion will be at least as strong as requiring that the linear span of a given collection of operators consists of hyponormal operators.(c) The notion will relate in a reasonable way to the questions pertaining to commuting normal extensions of commuting operators.In some ways, the present note continues the investigations started in [3], but in addition it paraphrases the statements of some known results and open problems in subnormal operator theory in terms of joint hyponormality.We begin by fixing some notation. The set of bounded linear operators on a Hubert space %? will be denoted by 3 §(%f), while J^'"1' will stand for the direct sum of ßf with itself m times. An mxm operator matrix sf = (Ay) will be the matrix with the operator Ay as the (ij)tb entry. The commutator AB -BA of two elements A and B in 3h'(ßf) will be denoted by [A, B]

On Subnormality and Formal Subnormality for Tuples of Unbounded Operators

“…The following Corollary is a generalization of Proposition 3 in [3]. The proof is an easy consequence of the previous Theorem 2.…”

“…According to the definition of a hyponormal multi-operator given by A. Athavale in the bounded case (see [3]) and not using the notion of commutativity, which is delicate for unbounded operators, we give a notion of joint hyponormality for unbounded operators :…”

“…If we are in the bounded case, this definition is equivalent to that of A. Athavale (see Remark 1 in [3]). Of course, the hyponormality is invariant under permutation, and all tuples consisting of elements of a jointly hyponormal multi-operator on D are also jointly hyponormal on D.…”

Disintegration of Measures and Contractive 2-Variable Weighted Shifts

“…[1], [8]). The n-tuple T is said to be normal if T is commuting and each T i is normal, and T is subnormal if T is the restriction of a normal n-tuple to a common invariant subspace.…”

Weak subnormality of operators

“…It is known ( [15 i, j=1 is positive, or equivalently, the (k + 1) × (k + 1) operator matrix in (0.4) is positive (via the operator version of Choleski's Algorithm), then the Bram-Halmos criterion can be rephrased as saying that T is subnormal if and only if T is k-hyponormal for every k м 1 ( [18]). Recall ( [1], [18], [5]) that T ∈ L (H ) is said to be weakly k-hyponormal if L S((T, T 2 , . .…”