Fuglede–Putnam type theorems for (p,k) $(p,k)$-quasihyponormal operators via hyponormal operators

Abstract

Abstract For Hilbert space operators S, X, and T, (S,X,T)∈FP $(S,X,T)\in FP$ means Fuglede–Putnam theorem holds for triplet (S,X,T) $(S,X,T)$, that is, SX=XT $SX=XT$ ensures S∗X=XT∗ $S^{\ast }X=XT^{\ast }$. Similarly, (S,T)∈FP $(S,T)\in FP$ means (S,X,T)∈FP $(S,X,T)\in FP$ holds for each operator X. This paper is devoted to the study of Fuglede–Putnam type theorems for (p,k) $(p,k)$-quasihyponormal operators via a class of operators based on hyponormal operators FP(H):={S|(S,T)∈FP holds for each hyponormal operator T∗} $FP(H):=\{S|(S,T)\in FP \mbox{ holds for each hyponormal operator } T^{\ast }\}$. Fuglede–Putnam type theorems involving (p,k) $(p,k)$-quasihyponormal, dominant, and w-hyponormal operators, which are extensions of the results by Tanahashi, Patel, Uchiyama, et al., are obtained.

Keywords

• Fuglede–Putnam theorem
• Hyponormal operator
• Dominant operator
• w-hyponormal operator