On the self-inverse operators

Abstract

Let T be an operator on a Hilbert space H. The problem of computing of the norm of T , norm of selfcommutators of T , and the numerical radius of T are discussed in many papers and a number of textbooks. In this paper we determine the relationships between these values for self inverse operators and explain how we can determine any three of these ( T , [T * , T ] , {T * , T } , and the numerical radius of T ) by knowing any one of them. Also, we find the spectrum of T * T , [T * , T ] and {T * , T } in the case that T is self inverse and the spectrum of T * T is an interval. Finally, by giving some examples on automorphic composition operators, we show that these results make it possible to replace lengthy computation with quick ones.