On a question of N. Salinas

Chō, Muneo

doi:10.1090/S0002-9939-1986-0848883-8

On a question of N. Salinas
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Proc. Amer. Math. Soc. 98 (1986), 94-96 Request permission

Abstract:

In [5], Salinas asked the following question: If $T = ({T_1}, \ldots ,{T_n})$ consists of commuting hyponormal operators, is it true that (1) $\delta (T - \lambda ) = d(\lambda ,{\sigma _\pi }(T))$ and (2) ${r_\pi }(T) = ||{D_T}||$? He proved that, for a doubly commuting $n$-tuple of quasinormal operators, (2) was true and (1) was true for $\lambda = 0$. In this paper we shall show that (2) holds for a doubly commuting $n$-tuple of hyponormal operators and give an example of a subnormal operator which does not satisfy (1).

References

Muneo Ch\B{o} and Makoto Takaguchi, Some classes of commuting $n$-tuples of operators, Studia Math. 80 (1984), no. 3, 245–259. MR 783993, DOI 10.4064/sm-80-3-245-259
Raul E. Curto, Fredholm and invertible $n$-tuples of operators. The deformation problem, Trans. Amer. Math. Soc. 266 (1981), no. 1, 129–159. MR 613789, DOI 10.1090/S0002-9947-1981-0613789-6
A. T. Dash, Joint numerical range, Glasnik Mat. Ser. III 7(27) (1972), 75–81 (English, with Serbo-Croatian summary). MR 324443
Takayuki Furuta, A counterexample to a result on approximate point spectra of polar factors of hyponormal operators, Yokohama Math. J. 32 (1984), no. 1-2, 225–226. MR 772918
Norberto Salinas, Quasitriangular extensions of $C^{\ast }$-algebras and problems on joint quasitriangularity of operators, J. Operator Theory 10 (1983), no. 1, 167–205. MR 715566
J. G. Stampfli, Which weighted shifts are subnormal?, Pacific J. Math. 17 (1966), 367–379. MR 193520, DOI 10.2140/pjm.1966.17.367
Joseph L. Taylor, A joint spectrum for several commuting operators, J. Functional Analysis 6 (1970), 172–191. MR 0268706, DOI 10.1016/0022-1236(70)90055-8