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 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.

Semiclosed projections and applications

“…Given an operator C : D(C) ⊆ H → H, the following are equivalent: The set SC(H) is closed under addition, multiplication, inversion and restriction to semiclosed subspaces of H, [28]. Also, if T 1 , T 2 ∈ SC(H, K) are such that T 1 and T 2 coincide on D(T 1 ) ∩ D(T 2 ), then the operator T : D(T 1 ) + D(T 2 ) → K coinciding with T 1 on D(T 1 ) and with T 2 on D(T 2) is a semiclosed operator, [21].…”