ON THE PUTNAM-FUGLEDE THEOREM OF NON-NORMAL OPERATORS

In this paper we have extended the Putnam-Fuglede Theorem of nomal operators and discussed the condition for the Putoam-Fuglede Theorem holding. We have proved that if A and $\[{B^*}\]$ are hyponomal operators and $\[AX = XB\]$, then $\[{A^*}X = X{B^*}\]$, that if A and $\[{B^*}\]$ are semi-hyponomal operators and X is invertible operator such that $\[AXB = X\]$, then \[{A^*}X{B^*} = X\], that if T is a contraction and P is a positive compact opertor such that $\[{T^*}PT = P\]$, then $\[\overline {R(P)} \]$ reduces T to unifary. In the meantime, we have proved that $\[AXB = X\]$ and $\[{A^*}X{B^*} = X\]$ both are true if and only if 1°$\[N{(X)^ \bot }\]$, $\[\overline {R(X)} \]$ reduce B, A to invertible operators, respectively; 2°Let $\[X = W{P_0}\]$ be polar decomposition, then we have that $\[{B^{ - 1}}{|_{N(X)}}\]$ and $\[A{|_{\overline {R(X)} }}\]$ are unitary equivalent by W which is unitary from $$\[N{(X)^ \bot }\]$$ to $\[R(X)\]$, and $\[{P_0}\]$ and B commute.