## 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

Zbl 0684.47020
Full Text:

### References:

 [1] BERBERIAN S. K., Conditions on an operator implying Re ?(T)=?(ReT), Trans. Amer. Math. Soc., Vol. 154 (1971) 267-272. · Zbl 0212.15702 [2] COWEN C., Hyponormal and subnormal Toeplitz operators, Surveys of some recent results in operator theory, Vol. I, Pitman Research Notes in Math. 171 (1988) 155-167. [3] DUNFORD N.-SCHWARTZ J. T., Linear operators, Part I, Interscience Publishers, New York (1958). · Zbl 0084.10402 [4] GOLDSTEIN J. A., Semigroups of linear operators and applications, Oxford Mathematical Monographs (1985). · Zbl 0592.47034 [5] JANAS J., On unbounded hyponormal operators, Arkiv för Math. Vol. 27 (1989) No. 2 273-281. · Zbl 0684.47020 · doi:10.1007/BF02386376 [6] JANAS J., Unbounded Toeplitz operators in the Bargmann-Segal space (preprint) (1989). · Zbl 0766.47004 [7] KATO T., Perturbation theory for linear operators, Springer-Verlag, Second Edition (1976). · Zbl 0342.47009 [8] LIZAMA C., On the spectrum of cosine operator function, Integral Equat. Operat. Th. Vol. 12 (1989) 714-723. · Zbl 0696.47044 [9] NAGY B., On cosine functions on Banach spaces, Acta Scient. Math. Szeged, 36 (1974) 281-290. · Zbl 0273.47008 [10] MARTIN M.-PUTINAR M., Lectures on hyponormal operators, Operator Theory, Vol. 39, Birkhäuser (1989). · Zbl 0684.47018 [11] SOVA M., Cosine operator functions, Rozprawy Mat. (1966) 1-46. · Zbl 0156.15404
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.