Shorted operators and minus order

TL;DR

This paper characterizes shorted operators in Hilbert spaces using the minus order and explores an operator approximation problem, showing the shorted operator as a minimal solution under certain conditions.

Contribution

It provides a new characterization of shorted operators via the minus order and links this to an operator approximation problem with explicit solutions.

Findings

Shorted operators are characterized as maxima and infima under the minus order.

The shorted operator to the range of A minimizes a specific operator approximation problem.

Conditions are identified under which the shorted operator is the minimal solution.

Abstract

Let $\mathcal{H}$ be a Hilbert space, $L(\mathcal{H})$ the algebra of bounded linear operators on $\mathcal{H}$ and $W \in L(\mathcal{H})$ a positive operator. Given a closed subspace $\mathcal{S}$ of $\mathcal{H}$, we characterize the shorted operator $W_{/ \mathcal{S}}$ of $W$ to $\mathcal{S}$ as the maximum and as the infimum of certain sets, for the minus order $\stackrel{-}{\leq}.$ Also, given $A \in L(\mathcal{H})$ with closed range, we study the following operator approximation problem considering the minus order: $$ min_{\stackrel{-}{\leq}} \ \{(AX-I)^*W(AX-I) : X \in L(\mathcal{H}), \mbox{ subject to } N(A^*W)\subseteq N(X) \}. $$ We show that, under certain conditions, the shorted operator $W_{/R(A)}$ (of $W$ to the range of $A$) is the minimum of this problem and we characterize the set of solutions.