Pairs of operators with trivial range intersection

Abstract

Let A and B be positive operators on a complex Hilbert space H. The parallel sum A: B of A and B is the strong operator limit (as ε ↓ 0) of the net ( A + ε)( A + B + 2 ε) −1( B + ε). If this net converges in norm, then we say that the pair ( A, B) is of class UUP. If Range A ⊆ Range( A + B), then ( A, B) is a UUP pair. The author has shown that if AB + BA + B 2 ⩾ 0, then Range A ⊆ Range( A + αB) for all real α with α ⩾ 1 and that if AB + BA ⩾ 0, then Range A ⊆ Range( A + αB) for all α > 0. In this paper we show that when Range( A) ∩ Range( B) = 0, then the condition that Range( A) + Range( B) should be closed is equivalent to several other sets of conditions involving the class UUP. We also prove that if T and K are operators on H satisfying Range( T) ∩ Range( K) = 0, and if A and B are the positive operators which appear in the polar decompositions of T and K, respectively, then Range( T) + Range( K) is closed if and only if A and B have closed ranges and ( A, B) is UUP.