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Operators of type \(\omega\) without a bounded \(H_{\infty}\) functional calculus. (English) Zbl 0709.47016
Miniconference on operators in analysis, Proc. Miniconf., Sydney/Australia 1989, Proc. Cent. Math. Anal. Aust. Natl. Univ. 24, 159-172 (1990).
[For the entire collection see Zbl 0699.00025.]
A. McIntosh [Proc. Cent. Math. Anal. Aust. Natl. Univ. 14, 210-231 (1990; Zbl 0634.47016)] gave several conditions equivalent to a closed operator of type \(\omega\) in a Hilbert space having a bounded \(H_{\infty}\) functional calculus.
The authors construct several examples of operators of type \(\omega\) which do not satisfy these conditions. Moreover, they show that every operator of type \(\omega\) does have a bounded \(H_{\infty}\) functional calculus if it is considered as acting in a Hilbert space obtained by a suitable renorming of the initial one.
Reviewer: M.Gonzalez

47A60 Functional calculus for linear operators