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**Operators which have an \(H_{\infty}\) functional calculus.**
*(English)*
Zbl 0634.47016

Operator theory and partial differential equations, Miniconf. Ryde/Aust. 1986, Proc. Cent. Math. Anal. Aust. Natl. Univ. 14, 210-231 (1986).

[For the entire collection see Zbl 0623.00012.]

From the introduction: An operator T in a Hilbert space H is said to be of type \(\omega\) if the spectrum is contained in the sector \(S_{\infty}=\{\zeta \in {\mathbb{C}}|| \arg \zeta | \leq \omega \}\) and the resolvent satisfies a bound of the type \(\| (T-\zeta I)^{-1}\| \leq C_{\mu}| \zeta |^{-1}\) for all \(\zeta\) with \(| \arg \zeta | \geq \mu\) and all \(\mu >\omega\). Let us suppose for now that T is one-one with dense range.

Such an operator has fractional powers T s and, if \(\omega <\pi /2\), generates an analytic semi-group \(\{\) exp(-tT)\(\}\). However it may or may not happen that it generates a C 0-group \(\{T^{is}|\) \(s\in {\mathbb{R}}\}\) of bounded operators. It was shown by Yagi that the operators T for which \(T^{is}\in L(H)\) are precisely those for which the domains of the fractional powers of T (and of T *) are the complex interpolation spaces between H and D(T) (and between H and D(T *)). They are also precisely those operators for which T and T * satisfy quadratic estimates [A. Yagi, C. R. Acad. Sci. Paris, Sér. I 299, 173-176 (1984; Zbl 0563.46042)].

It is shown in this paper that another equivalent property is the existence of an \(H_{\infty}(S\) \(0_{\mu})\) functional calculus for \(\mu >\omega\) (where \(S\) \(0_{\mu}\) denotes the interior of \(S_{\mu}).\)

The material in this paper has two heritages. One is operator theory, the other is harmonic analysis, where the power of quadratic estimates has been recognized since the Littlewood-Paley theory appeared and the theory of g-functions was developed by Zygmund and his followers. The motivation for this paper is to better understand the functional calculus of \(i^{- 1}d/dz|_{\gamma}\), where \(\gamma\) is a Lipschitz curve in the complex plane, though in fact this material is only briefly alluded to in the last section. This builds upon the work of Calderón, and of Coifman and Meyer.

From the introduction: An operator T in a Hilbert space H is said to be of type \(\omega\) if the spectrum is contained in the sector \(S_{\infty}=\{\zeta \in {\mathbb{C}}|| \arg \zeta | \leq \omega \}\) and the resolvent satisfies a bound of the type \(\| (T-\zeta I)^{-1}\| \leq C_{\mu}| \zeta |^{-1}\) for all \(\zeta\) with \(| \arg \zeta | \geq \mu\) and all \(\mu >\omega\). Let us suppose for now that T is one-one with dense range.

Such an operator has fractional powers T s and, if \(\omega <\pi /2\), generates an analytic semi-group \(\{\) exp(-tT)\(\}\). However it may or may not happen that it generates a C 0-group \(\{T^{is}|\) \(s\in {\mathbb{R}}\}\) of bounded operators. It was shown by Yagi that the operators T for which \(T^{is}\in L(H)\) are precisely those for which the domains of the fractional powers of T (and of T *) are the complex interpolation spaces between H and D(T) (and between H and D(T *)). They are also precisely those operators for which T and T * satisfy quadratic estimates [A. Yagi, C. R. Acad. Sci. Paris, Sér. I 299, 173-176 (1984; Zbl 0563.46042)].

It is shown in this paper that another equivalent property is the existence of an \(H_{\infty}(S\) \(0_{\mu})\) functional calculus for \(\mu >\omega\) (where \(S\) \(0_{\mu}\) denotes the interior of \(S_{\mu}).\)

The material in this paper has two heritages. One is operator theory, the other is harmonic analysis, where the power of quadratic estimates has been recognized since the Littlewood-Paley theory appeared and the theory of g-functions was developed by Zygmund and his followers. The motivation for this paper is to better understand the functional calculus of \(i^{- 1}d/dz|_{\gamma}\), where \(\gamma\) is a Lipschitz curve in the complex plane, though in fact this material is only briefly alluded to in the last section. This builds upon the work of Calderón, and of Coifman and Meyer.

Reviewer: A.McIntosh