Abstract
Let T be an operator on a Hilbert space \( \mathcal{H} \). 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.
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Research partially supported by the Shiraz university Research Council Grant No. 86-GR-SC-32.
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Abdollahi, A. On the self-inverse operators. Rend. Circ. Mat. Palermo 58, 257–264 (2009). https://doi.org/10.1007/s12215-009-0019-x
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DOI: https://doi.org/10.1007/s12215-009-0019-x