Almost regular operators are regular (Q2712556)

Let \(X\) and \(Y\) be real or complex Banach spaces and let \(L(X,Y)\) denotes the set of all bounded linear operators acting from \(X\) into \(Y\). An operator \(T\in L(X,Y)\) is said to be almost regular if there is a bounded sequence \(\{A_n\}\subset L(X,Y)\) such that \(\|TA_n T- T\|\to 0\) as \(n\to\infty\). \(T\in L(X,Y)\) is said to be regular if there is an \(A\in L(X,Y)\) such that \(TAT= T\). By definition, if \(T\) is regular then it is almost regular. Main result: \(T\) is almost regular if and only if \(T\) is regular.