Complex Jordan operations
type: Item
instance of: scholarly article
title: Complex Jordan operations
zbMATH Open document ID: 0532.47024
DOI: 10.1007/BF01691926
author: John W. Bunce
published in: Integral Equations and Operator Theory
publication date: 1983-01-01T00:00:00Z
review text: An operator J on a Hilbert space is called a complex Jordan operator if \(J=M+N\) where M is normal, \(N^ 2=0\) and \(MN=NM.\) If M is selfadjoint, then J is a Jordan operator. \textit{J. W. Helton} [Trans. Am. Math. Soc. 170, 305-331 (1972; Zbl 0251.47031)] proved that J is Jordan if and only if it satisfies the two equations \(J^{*^ 3}-3J^{*^ 2}J+3J^*J^ 2-J^ 3=0\) and \(J^ 3-3J^ 2J^*+3JJ^{*^ 2}-J^{*^ 3}=0.\) In the paper under review the author gives an algebraic characterization of complex Jordan operators: An operator J is complex Jordan if and only if it satisfies the three equations \[ J^{*^ n}J^3 - 3JJ^{*^ n}J^2 + 3J^2J^{*^ n}J - J^3 J^{*^ n} = 0, \quad n=1,2\text{ and }3. \] The result is applied to give another proof of Helton's characterization of Jordan operators. The author also discusses the independence of the three equations in his characterization of complex Jordan operators.
Mathematics Subject Classification ID: 47B40
Mathematics Subject Classification ID: 47B15
zbMATH DE Number: 3844443
zbMATH Keywords: normal operator
zbMATH Keywords: binormal operator
zbMATH Keywords: complex Jordan operator
reviewed by: Katsutoshi Takahashi
MaRDI profile type: Publication
cites work: Q3969420
cites work: Nonnormal dilations, disconjugacy and constrained spectral factorization
cites work: The Unitary Equivalence of Binormal Operators
cites work: Completely positive maps on C*-algebras and the left matricial spectra of an operator
cites work: Q4403351
cites work: Infinite Dimensional Jordan Operators and Sturm-Liouville Conjugate Point Theory
description: scientific article; zbMATH DE number 3844443