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

*(English)* Zbl 1219.47036
Summary: An operator $T\in B\left(H\right)$ is called quasi-class $(A,k)$ if ${T}^{*k}\left(\right|{T}^{2}|-|{T}^{2}\left|\right){T}^{k}\ge 0$ for a positive integer $k$, which is a common generalization of class A. The famous Fuglede-Putnam theorem is as follows: the operator equation $AX=XB$ implies ${A}^{*}X=X{B}^{*}$ 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 ${B}^{*}$ is an invertible class A operator such that $AX=XB$, then ${A}^{*}X=X{B}^{*}$. Secondly, we consider 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.

##### MSC:

47B20 | Subnormal operators, hyponormal operators, etc. |

47A63 | Operator inequalities |