Shorted operators and minus order

Abstract

Let H be a Hilbert space, L(H) the algebra of bounded linear operators on H and W ∈ L(H) a positive operator. Given a closed subspace S of H, we characterize the shorted operator W /S of W to S as the maximum and as the infimum of certain sets, for the minus order − ≤. Also, given A ∈ L(H) with closed range, we study the following operator approximation problem considering the minus order:We show that, under certain conditions, the shorted operator of W /R(A) is the minimum of this problem and we characterize the set of solutions. KEYWORDS Shorted operators; minus order; oblique projections ≤ B (where the symbol − ≤ stands for the minus order of operators) if the Dixmier angle between R(A) and R(B − A) and the Dixmier angle between R(A * ) and R(B * − A * ) are less than 1, where R(T ) stands for the range of the operator T. Independently, in [31]Šemrl gave another characterization of the minus order, he showed that A − ≤ B if and only if there are bounded (oblique) projections, i.e. idempotents, P and Q such that A = P B and A * = QB * . In [18], a new characterization of the minus order for operators acting on Hilbert spaces was given in terms of the so called range additivity property. Namely, it was proved that A − ≤ B is equivalent to the range of B being the direct sum of ranges of A and B − A and the range of B * being the direct sum of ranges of A * and B * − A * , which generalizes previous results presented in [31]. This plays an equivalent role to the rank additivity characterization when A and B are matrices [21, 26]. In [5] the notion of shorted operator appears in relation with the minus order. Given a closed subspace S of H and W ∈ L(H) a positive operator, in 1947, Krein [23], proved the existence of a maximum (with respect to the order induced by the cone of positive operators) of the set M(W, S) = {X ∈ L(H) : 0 ≤ X ≤ W and R(X) ⊆ S ⊥ }.Krein used this extremal operator in his theory of extension of symmetric operators. Years later, Anderson and Trapp [2] studying the same problem, called this maximum the shorted operator of W to S (in the following denoted W /S ) and showed interesting properties of this operator and its connections with electrical circuit theory. The shorted operator have shown to be useful in many applications [5], [32].The pair (W, S) is said to be compatible if there exists a bounded linear (not necessarily selfadjoint) projection Q onto S such that W Q is selfadjoint. Thus, ifthen (W, S) is compatible if and only if P(W, S) is not empty. In [15], it was shown that there exists a strong relationship between compatibility, the projections of P(W, S) and the shorted operator W /S . Later, in [5], the notion of shorted operator was generalized to that of bilateral shorted operator, for an operator W ∈ L(H) (not necessarly positive) and a pair of closed subspaces. The proposed definition comes from the notion of weak complementability, which is a refinement of a finite dimensional notion due to T. Ando [3]. In that paper, it was proven that given an operator W ∈ L(H) positive an...