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]

Seminormal systems of operators in Clifford environments

“…Apparently the two other classes should be either ruled out, or regarded as pathological. This would be wrong, and the following theorem due to Athavale [1], as well as some of our subsequent results will make the point.…”

“…Prompted by this result we introduce two new definitions. For the benefit of our reader, we should mention that the concepts of joint hyponormality and joint t-hyponormality introduced by Athavale [1] and Xia [44], and eventually adopted and studied by other researchears, correspond in our terminology to joint left hyponormality and joint right hyponormality, respectively. To get a common ground, we just need to observe that according to Lemma 3.6, each of the self-commutator operator matrices defined by equations (3.10) and (3.11) is derived as a compression of a certain specific self-commutator operator form.…”

“…Our goal is to single out and motivate what we believe to be the most natural counterparts of the previous equations and requirements in a multidimensional setting, i.e., for systems of operators. Significant early contributions to the development of the theory of joint seminormality, which prompted us to try and find a unifying framework, are due to Athavale [1], Cho, Curto, Huruya, and Zelazko [7], Curto [11], Curto and Jian [12], Curto, Muhly, and Xia [13], Douglas, Paulsen, and Yan [14], Martin and Salinas [29], McCullough and Paulsen [31], and Xia [44]. The viewpoint emphasized in our article was already outlined in Martin [19,[22][23][24] and is based on some techniques from Clifford analysis and spin geometry.…”

An answer to a question of A. Lubin: The lifting problem for commuting subnormals

“…Joint subnormality. The notion of joint hyponormality for the general case of ntuples of operators was first formally introduced by A. Athavale [5]. Joint hyponormality originated from the LPCS, and it has also been considered with an aim at understanding the gap between hyponormality and subnormality for single operators.…”

“…By analogy with the case n = 1, we shall say ( [5], [12]) that T is jointly hyponormal (or simply, hyponormal) if [T * , T] is a positive operator on H ⊕ · · · ⊕ H. Thus, the Bram-Halmos criterion can be restated as: T ∈ B(H) is subnormal if and only if (T, T 2 , · · · , T k ) is hyponormal for every k ∈ Z + . The n-tuple T ≡ (T 1 , .…”