ON THE PUTNAM-FUGLEDE THEOREM OF NON-NORMAL OPERATORS

In this paper we have extended the Putnam-Fuglede Theorem of nomal operators anddiscussed the condition for the Putnam-Fuglede Theorem holding.We have proved that ifA and B~* are hyponomal operators and AX=XB,then A~*X=XB~*;that if A and B~* aresemi-hyponomal operators and X is

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 ifA 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 unitaryequivalent by W which is unitary from $$[N{(X)^ bot }]$$ to $[R(X)]$, and $[{P_0}]$ and B commute.