Solvability of The Operator Equations $ AX-XB=C $ and $ AX-YB=C $

This paper provides new necessary and sufficient conditions for the solvability to the operator equations $AX-XB=C$ and $AX-YB=C,$ where $A$ and $B$ are group invertible operators defined on an infinite dimensional Hilbert space. In addition, the general solutions to the equation $AX-YB=C,$ are derived in terms of group inverse of $A$ and $B$. As a consequence, new necessary and sufficient conditions for the solvability to the operator equation $AYB-Y=C,$ are derived.