  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.
