The range of A−1A∗ in G L( n,C)

Abstract

Let A be an invertible operator on a finite dimensional complex Hilbert space. We carry out a detailed study of the map A→A −1A ∗≡Φ(A) . It is shown that the range of Φ is exactly the set of all invertible operators T for which T −1 is similar to T ∗. In particular, unitaries and similarities of unitaries are in the range of Φ and we prove, among other things, the equivalence of the assertions: (i) T is similar to a unitary, (ii) every A ∈ Φ −1( T) is congruent to a normal operator, (iii) there exists B ∈ Φ −1( T) whose field of values omits the origin of the complex plane. For general T in the range of Φ, we determine all A ∈ Φ −1( T) in terms of the self-adjoint invertible operators fixed by the map X→ T ∗ XT. Many of the results contained in this paper have known analogues for operators which are similar to their adjoints.