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arXiv · 1802.01973

Shorted operators and minus order

Maximiliano Contino·Juan Ignacio Giribet·Alejandra Maestripieri
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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.

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Maximiliano Contino, Juan Ignacio Giribet, Alejandra Maestripieri. 2018-02-06. Shorted operators and minus order. https://arxiv.org/abs/1802.01973

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