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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.

In this paper, we study two systems of generalized Sylvester operator equations. We derive necessary and sufficient conditions for the existence of a solution and provide the general form of a solution. We extend some recent results to more general settings.