Abstract
The paper is concerned with the following question: if A and B are two bounded operators between Hilbert spaces H and K, and M and N are two closed subspaces in H, when will there exist a bounded operator C:H→K which coincides with A on M and with B on 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.
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