On some properties of the additivity of operators ranges

M. L. ARIAS; G. CORACH; A. MAESTRIPIERI

Encuentro; AMS-EMS-SPM International Meeting 2015; 2015

Given two bounded linear operators on a Hilbert space H, we say that they satisfy the  range additivity property if $R(A+B)=R(A)+R(B),$ where R(T) denotes the range of T. In this talk we shall present some results concerning this property and its relationship with the notion of compatiblity and shorted operator. Given a positive operator A in L(H)^+ and a closed subspace S included in H we say that they are  compatible if S+(AS)^\bot=H. This is equivalent to the existence of an idempotent operator with range S and selfadjoint respect to the semi-inner product induced by A. On the other hand, the  shorted operator, defined as [S]A := max {X in L(H)^+ : X < A and  R(X) is included in S}, was introduced by M. G. Krein to explore the selfadjoint extensions of operators. Later, W. N. Anderson and G. E. Trapp redefined it and used it in the study of electrical networks.The relationship between these two last notions was studied by G. Corach, A. Maestripieri and D. Stojanoff. Here, we explore how these concepts are also related with the range additivity property and we show some applications to certain operator factorizations.