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On unbounded hyponormal operators. II. (English) Zbl 0772.47005

Let \(T\) be a densely defined operator on a complex Hilbert space. Then \(T\) is called hyponormal, if \(D(T)\subseteq D(T^*)\) and \(\| T^* x\|\leq \| Tx\|\), \(\forall x\in D(T)\). This extension of the concept of hyponormal operators to the case of unbounded operators was given by the author in a previous publication (part I), where he studied some basic properties of such operators [see J. Janas, Ark. Mat. 27, No. 2, 273-281 (1989; Zbl 0684.47020)].
The present paper continues this study, providing some interesting examples of unbounded hyponormal operators. The author shows also that if \(A\) is a closed hyponormal operator with spectrum in the parabola \(\{(x,y)\); \(x\geq by^ 2\}\), then \(-A\) generates a cosine operator family. The proof is based on the fact that \(A(A+t)^{-1}\), \(t>0\), is bounded and hyponormal, which implies that \(A^{1/2}\) is \(m\)-accretive with numerical range in the half strip \(\{(x,y)\): \(x\geq 0\), \(| y|\leq 1/2b^{1/2}\}\).

MSC:

47B20 Subnormal operators, hyponormal operators, etc.
47D09 Operator sine and cosine functions and higher-order Cauchy problems
47B44 Linear accretive operators, dissipative operators, etc.
47A12 Numerical range, numerical radius

Citations:

Zbl 0684.47020
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References:

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