On p-hyponormal operators for \(0<p<1\). (English) Zbl 0718.47015

A Hilbert space operator T is called p-hyponormal if \((T^*T)^ p- (TT^*)^ p\geq 0\) for some \(p>0\). Several properties of the polar decompositions of such operators are studied. For example if \(T=U| T|\) is the polar decomposition, then \(r_{\sigma}(T)=\| T\|\) and every eigenvalue of U reduces T provided that U is unitary and \(0<p<\). A similar result is true for eigenvalues of \(| T|\) if moreover \(\sigma\) (U) is not the whole unit circle. For invertible T with \(0<p<\) it is shown that every \(\lambda \in \sigma_ p(T)\) can be decomposed as \(\lambda e^{i\theta}\) where \(\lambda \in \sigma_ p(| T|)\), \(e^{i\theta}\in \sigma_ p(U)\), and \((| T| - \lambda)x=(U-e^{i\theta})x=0\) for some \(x\neq 0\). For \(0<p<1\) and T invertible, the growth condition \(\| | T|^{1/2}(T- \lambda)^{-1}| T|^{-1/2}\| =1/dist(\lambda,\sigma (T))\) for \(\lambda\not\in \sigma (T)\) is proved.


47B20 Subnormal operators, hyponormal operators, etc.
47A10 Spectrum, resolvent
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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