Annals of Functional Analysis

The Fuglede-Putnam theorem and Putnam's inequality for‎ ‎quasi-class $(A‎, ‎k)$ operators

Abstract

‎An operator $T \in B(\mathcal{H})$ is called quasi-class $(A‎, ‎k)$ if $T^{\ast‎ ‎k}(|T^{2}|-|T|^{2})T^{k} \geq 0$ for a positive integer $k$‎, ‎which‎ ‎is a common generalization of class A‎. ‎The famous Fuglede-Putnam's‎ ‎theorem is as follows‎: ‎the operator equation $AX=XB$ implies‎ ‎$A^{\ast}X=XB^{\ast}$ when $A$ and $B$ are normal operators‎. ‎In this‎ ‎paper‎, ‎firstly we show that if $X$ is a Hilbert-Schmidt operator‎, ‎$A$ is a quasi-class $(A‎, ‎k)$ operator and $B^{\ast}$ is an‎ ‎invertible class A operator such that $AX=XB$‎, ‎then‎ ‎$A^{\ast}X=XB^{\ast}$‎. ‎Secondly we consider the Putnam's inequality‎ ‎for quasi-class $(A‎, ‎k)$ operators and we also show that‎ ‎quasisimilar quasi-class $(A‎, ‎k)$ operators have equal spectrum and‎ ‎essential spectrum‎.

Article information

Source
Ann. Funct. Anal., Volume 2, Number 1 (2011), 105-113.

Dates
First available in Project Euclid: 12 May 2014

https://projecteuclid.org/euclid.afa/1399900266

Digital Object Identifier
doi:10.15352/afa/1399900266

Mathematical Reviews number (MathSciNet)
MR2811211

Zentralblatt MATH identifier
1219.47036

Subjects
Primary: 47A63: Operator inequalities
Secondary: 47B20: Subnormal operators, hyponormal operators, etc.

Citation

Gao, Fugen; Fang, Xiaochun. The Fuglede-Putnam theorem and Putnam's inequality for‎ ‎quasi-class $(A‎, ‎k)$ operators. Ann. Funct. Anal. 2 (2011), no. 1, 105--113. doi:10.15352/afa/1399900266. https://projecteuclid.org/euclid.afa/1399900266

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