Generalized Fuglede-Putnam Theorem and $m$-quasi-class $A(k)$ operators
Abstract
For a bounded linear operator $T$ acting on a
complex infinite dimensional Hilbert space $\h,$ we say that $T$
is $m$-quasi-class $A(k)$ operator for $k>0$ and $m$ is a
positive integer (abbreviation $T\in\QAkm$) if
$T^{*m}\left((T^*|T|^{2k}T)^{\frac{1}{k+1}}-|T|^2\right)T^m\geq
0.$ The famous {\it Fuglede-Putnam theorem} asserts that: the operator equation
$AX=XB$ implies $A^*X=XB^*$ when $A$ and $B$ are normal operators.
In this paper, we prove that if $T\in \QAkm$ and $S^*$ is
an operator of class $A(k)$ for $k>0$. Then $TX=XS$, where $X\in
\bh$ is an injective with dense range implies $XT^*=S^*X$.
Keywords
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DOI: https://doi.org/10.22190/FUMI1901073R
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