On the spectrum of \(p\)-hyponormal operators. (English) Zbl 0893.47013

Duggal, B. P.

Acta Sci. Math. 63, No. 3-4, 623-637 (1997).

A bounded linear operator \(A\) on a Hilbert space is called to be \(p\)-hyponormal for \(0<p\leq 1\) if \((AA^*)^p\leq (A^*A)^p\). Results known for hyponormal operators are shown to be valid for \(p\)-hyponormal operators, too, such as properties relating the spectra of \(A,| A|,A^*,| A^*|, | A^*|^2\); Weyl’s theorem is proven for \(f(A)\) if \(f\) is a function analytic in a neighbourhood of \(\sigma(A)\); and Putnam’s estimation of the norm of the self-commutator by the measure of the spectrum is generalized. A main tool is to consider the operator \(|\widehat A|^{\frac 12} V| \widehat A|^{\frac 12}\) where the operator \(\widehat A\) is defined by \(\widehat A:=| A|^{\frac 12} U | A|^{\frac 12}\) with polar decomposition \(\widehat A=V| \widehat A|\) and \(U\) is the partial isometry from the polar decomposition \(A=U| A|\).

Reviewer: Gerhard Garske (Hagen)

MSC:

47B20	Subnormal operators, hyponormal operators, etc.
47A10	Spectrum, resolvent
47A60	Functional calculus for linear operators

Keywords:

hyponormal operators; spectrum; Weyl’s theorem; Putnam’s estimation of the norm of the selfcommutator; isometry; polar decomposition