Measure, Integration & Real Analysis · Problem 10A.14
Linear Maps on Hilbert Spaces — Sheldon Axler
Mathematics — Measure Theory · unique-left-right-inverse
Problem
Suppose T is a bounded operator on a Hilbert space V.
(a) Prove that T is invertible if and only if T has a unique left inverse. In other words, prove that T is invertible if and only if there exists a unique S∈B(V) such that ST=I.
(b) Prove that T is invertible if and only if T has a unique right inverse. In other words, prove that T is invertible if and only if there exists a unique S∈B(V) such that TS=I.