Some equivalent conditions of stable perturbation of operators in Hilbert spaces
A number of results are presented concerning the perturbation of a bounded operator between Hilbert spaces \(H_1\) and \(H_2\) which has a Moore-Penrose (MP) generalized inverse. Let \(T_0\in B(H_1,H_2)\) with MP inverse \(T^+_0\) and let \(T= T_0+\delta T\in B(H_1, H_2)\). If \(R(T)\cap R(T_0)^\perp= 0\) then \(T\) is said to be a stable perturbation of \(T_0\). Five equivalent conditions are given that \(T\) be a stable perturbation of \(T_0\) given that \(\| T^+_0\|\,\|\delta T\|< 1\). If \(R(T)\cap R(T_0)^\perp= 0\), \(\| T^+_0\|\,\|\delta T\|< 1\) then \(T^+\) satisfies \[ \| T^+- T^+_0\|\leq {1+\sqrt{5}\over 2} \| T^+\|\,\| T^+_0\|\,\|\delta T\|. \] The authors find the optimal solution to the perturbed orthogonal projection problem: Find \(y\in H_1\) such that \(\| p+\delta p- y\|\) is minimized, subject to the condition \[ \| Ty- b-\delta b\|= \min_{z\in H}\,\| Tz- b-\delta b\|, \] \(p,\delta p\in H _1\), \(b,\delta b\in H_2\).
scientific article; zbMATH DE number 2009138