Simultaneous extension of two bounded operators between Hilbert spaces

Abstract

The paper is concerned with the following question: if $A$ and $B$ are two bounded operators between Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, and $\mathcal{M}$ and $\mathcal{N}$ are two closed subspaces in $\mathcal{H}$, when will there exist a bounded operator $C:\mathcal{H}\to\mathcal{K}$ which coincides with $A$ on $\mathcal{M}$ and with $B$ on $\mathcal{N}$ simultaneously? Besides answering this and some related questions, we also wish to emphasize the role played by the class of so-called semiclosed operators and the unbounded Moore-Penrose inverse in this work. Finally, we will relate our results to several well-known concepts, such as the operator equation $XA=B$ and the theorem of Douglas, Halmos' two projections theorem, and Drazin's star partial order.