Unilaterally invertible normals are invertible (Q2730928)

Let \(A\) be a complex unital Banach algebra with unit. An element of \(A\) is hermitian if it has real spatial numerical range when considered as an operator in \(\mathcal L(A)\). An element \(r\) is called normal if there are two hermitian elements \(s, t\) of \(A\) such that \(st = ts\) and \(r = s + it.\) It is shown that every normal element of \(A\) is unilaterally invertible.