Operators satisfying a similarity condition

ABSTRACTGiven Hilbert space operators A,S∈B(H) such that 0∉W(S)¯ (= the closure of the numerical range of S), the similarities ASA∗=S for invertible A and AS=SA∗ have been considered by a number of authors over past few decades. A classical result of C. R. De Prima (resp., I. H. Sheth) says that if A and A-1 are normaloid or convexoid (resp., A is hyponormal), then ASA∗=S implies A is unitary (resp., AS=SA∗ implies A is self-adjoint). This paper uses (Putnam–Fuglede theorem type) commutativity results to obtain generalizations of extant results on similarities of the above type. Amongst other results, it is proved that if ASA∗=S with A invertible and 0∉W(S)¯ , then: (i) A normaloid implies either A is unitary or σp(A)=∅ ; (ii) operators A satisfying the positivity condition |A2|2-2|A|2+I≥0 are unitary. If the operator A in ASA∗=S (resp., AS=SA∗ ) is w-hyponormal or class A(1,1) with A-1(0)⊆A∗-1(0) , then a sufficient condition for A to be unitary (resp., A to be self-adjoint) is that 0∉W(X) ; furthermore,...