Fuglede–Putnam theorem for class $A$ operators

Salah Mecheri¹

¹ Department of Mathematics College of Sciences Taibah University Medina, Saudi Arabia

Colloquium Mathematicum, Tome 138 (2015) no. 2, pp. 183-191.

Résumé

Let $A\in B (H)$ and $B\in B(K)$. We say that $A$ and $B$ satisfy the Fuglede–Putnam theorem if $AX = XB$ for some $X\in B(K,H)$ implies $A^{*}X = XB^{*}$. Patel et al. (2006) showed that the Fuglede–Putnam theorem holds for class $A(s,t)$ operators with $s+t 1$ and they mentioned that the case $s=t=1$ is still an open problem. In the present article we give a partial positive answer to this problem. We show that if $A\in B(H)$ is a class $A$ operator with reducing kernel and $B^{*}\in B(K)$ is a class $\mathcal {Y}$ operator, and $AX=XB$ for some $X\in B(K,H)$, then $A^{*}X=XB^{*}$.

Détail

DOI : 10.4064/cm138-2-3

Keywords: say satisfy fuglede putnam theorem implies * * patel showed fuglede putnam theorem holds class operators mentioned still problem present article partial positive answer problem class operator reducing kernel * class mathcal operator * *