A characterization of group generators on Hilbert spaces and the \(H^\infty\) -calculus
The author defines a functional calculus for operators with spectrum in a vertical strip by transforming the technique originally developed by Alan McIntosh for sectorial operators [see \textit{A. McIntosh}, Proc. Cent. Math. Anal. Aust. Natl. Univ. 14, 210--231 (1986; Zbl 0634.47016)]. In a Hilbert space, the operator \(A\) has certain ``quadratic estimates'' if and only if \(A\) generates a \(C_0\)-group of operators. Equivalently, the functional calculus has the \(H^\infty\)-property \[ \| f(A)\|\leq K\| f\|_\infty \text{ with \(K\) a constant.} \] An \(H^\infty\)-calculus for generators \(A\) of \(C_0\)-groups of Hilbert space operators \(e^{tA}\) was introduced ten years ago by the reviewer and \textit{R. de Laubenfels} [Proc. Am. Math. Soc. 120, 127--136 (1994; Zbl 0820.47017)]; see also \textit{K. N. Boyadzhiev} [Lect. Notes Pure Appl. Math. 175, 97--104 (1996; Zbl 0842.47010)]. The Boyadzhiev-de Laubenfels calculus uses Poisson's kernel for the strip. The estimate obtained there (the constant \(K\) in the above inequality) has the property that \(K=1\) when \(e^{tA}\) is a group of isometrics, i.e., in this case \(\| f(A)\| \leq \| f \|_\infty\). This makes it possible to obtain Stone's theorem for such groups as a corollary. The estimate given in the present paper does not have this property. About the role of quadratic estimates in the generation of \(C_0\)-groups, see also \textit{K. N. Boyadzhiev} [Collect. Math. 45, 287--300 (1994; Zbl 0819.47018)] and the recent article by \textit{M. Malejki} [Semigroup Forum 63, 305--320 (2001; Zbl 1034.47014)].
operators with spectrum in a vertical strip
scientific article; zbMATH DE number 1889243