A unitary invariant for hyponormal operators
type: Item
instance of: scholarly article
title: A unitary invariant for hyponormal operators
zbMATH Open document ID: 0635.47023
DOI: 10.1016/0022-1236(87)90070-X
author: Mircea Martin
author: Mihai Putinar
published in: Journal of Functional Analysis
publication date: 1987-01-01T00:00:00Z
review text: Let \({\mathcal H}\) be a complex separable Hilbert space, let \({\mathcal L}({\mathcal H})\) denote the algebra of all bounded linear operators on \({\mathcal H}\), and let T denote a hyponormal operator on \({\mathcal H}\) (i.e. \(D:=T\) *T-TT *\(\geq 0)\). Then for any \(z\in {\mathbb{C}}\) in \({\mathcal L}({\mathcal H})\) there exists a uniquely defined contraction C(z) in \({\mathcal L}({\mathcal H})\) such that (T-z) \(*C(z)=D^{1/2}\) and C *(z)\(| Ker(T-z)\) \(*=0\). This family of contractions is the main tool in this nicely readable paper. Generalizing a result of \textit{K. F. Clancey}'s [J. Oper. Theory 13, 265- 289 (1985; Zbl 0588.47043)] it is shown that T is pure (i.e. there exists no reducing subspace on which the restriction is normal) if and only if the span of all vectors C(z)x with z in \({\mathbb{C}}\) and x in \({\mathcal X}:=Ran D^{1/2}\) is dense in \({\mathcal H}\). The germ at infinity of the \({\mathcal L}({\mathcal H})\)-valued function C *C as well as the distribution \(- D^{1/2}\partial C\) in \({\mathcal D}'({\mathbb{C}}){\hat \otimes}{\mathcal L}({\mathcal X})\) (\(\partial C\) is the complex derivative of C in the sense of distributions) are exhibited to be complete unitary invariants. Originated by \textit{Xia}'s first representation of hyponormal operators as singular integral operators [Chinese J. Math. 3, 232-246 (1963)] a two- dimensional singular-integral model is developed. Finally those positive definite kernels in \({\mathcal L}^{\infty}({\mathbb{C}}\) 2)\({\hat \otimes}_{\epsilon}{\mathcal L}({\mathcal X})\) are characterized which produce pure hyponormal operators.
Mathematics Subject Classification ID: 47B20
zbMATH DE Number: 4034459
zbMATH Keywords: family of contractions
zbMATH Keywords: no reducing subspace
zbMATH Keywords: complete unitary invariants
zbMATH Keywords: two-dimensional singular-integral model
zbMATH Keywords: positive definite kernels
zbMATH Keywords: pure hyponormal operators
reviewed by: Gerhard Garske
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4034459