Boyadzhiev, Khristo N.; deLaubenfels, Ralph J. \(H^ \infty\) functional calculus for perturbations of generators of holomorphic semigroups. (English) Zbl 0738.47018 Houston J. Math. 17, No. 1, 131-147 (1991). The authors introduce and describe the properties of two functional calculi. The first one is an extension of the functional calculus introduced by the second author [Houston J. Math. 13, 545-548 (1987; Zbl 0653.47008)]. Due to the reduced hypoteses considered in the paper under review, the results may be applied to, for example, \(\epsilon + \Delta\), on \(L^ p(R^ m)\), for \(1\leq p<\infty\). This is not possible using the approach given in [loc. cit.].Let \(A\) be a closed, densely defined operator on the complex Banach space \(X\) with spectrum and numerical range contained in the set \(\epsilon+S(\theta)\), where \(\epsilon>0\) and \(S(\theta)\) is the sector \(\{z\in \mathbb{C}\mid\;\arg(z)\leq\theta<\pi/2\}\) so that, in particular, \(- (A-\epsilon)\) is the generator of a holomorphic semigroup of contractions \(e^{-z(A-\epsilon)}\) with angle \(\pi/2-\theta\) [A. Pazy, Semigroups of linear operators and applications to partial differential equations (1983; Zbl 0516.47023)]. Let \(B\) be a bounded Hermitian operator on \(X\) (i.e. \(e^{itB}\), \(t\in \mathbb{R}\), is a one-parameter group of isometries) commuting with \(e^{-tA}\) for all \(t\in \mathbb{R}\). Denoting by \(D(A)\) the domain of \(A\) and by \(B(X)\) the space of all bounded linear operators from \(X\) into itself, the authors define the functional calculus \(f\to\overline {f(A+iB)}\) by the formula \[ f(A+iB)x={1 \over \pi}\int_{-\infty}^{+\infty}f(-ir)(A^ 2+(r+B)^ 2)^{-1}Ax dr, \quad x\in D(A), \] for all functions \(f\in H_ 0^ \infty (RHP)\equiv\{\underline f \) holomorphic on \(\{z\mid\hbox{Re}(z)>0\}\) continuous on \(\{z\mid\hbox{Re}(z)\geq 0\}\) and bounded \(\}\).The second functional calculus, \(f\to\overline {f(BA)}\), is defined for a subclass of \(H_ 0^ \infty(RHP)\) as follows: \[ f(BA)x={1 \over \pi}\int_{-\infty}^{+\infty} f(-irB)(A^ 2+r^ 2)^{-1}Ax dr, \quad x\in D(A). \] A very interesting result of this paper is that although even on a Hilbert space, \(f(A)\) may not be bounded for all \(\underline f\) in \(H_ 0^ \infty(RHP)\) and \(A\) an operator of type \(\theta\) [A. McIntosh, Cent. Math. Anal. Aust. Natl. Univ. 14, 210- 231 (1986; Zbl 0634.47016)], \(f(A+iB)A^{-1}\) is bounded for such \(\underline f\). The authors show that this may be considered as an uniform control over the unboundedness, which, perhaps contrary to intuition, will in general exist. Reviewer: D.Savin (Montreal) Cited in 3 Documents MSC: 47A60 Functional calculus for linear operators 46H30 Functional calculus in topological algebras 47D06 One-parameter semigroups and linear evolution equations Keywords:closed densely defined operator; numerical range; holomorphic semigroup of contractions; bounded Hermitian operator; one-parameter group of isometries; functional calculus; uniform control over the unboundedness Citations:Zbl 0653.47008; Zbl 0516.47023; Zbl 0634.47016 PDFBibTeX XMLCite \textit{K. N. Boyadzhiev} and \textit{R. J. deLaubenfels}, Houston J. Math. 17, No. 1, 131--147 (1991; Zbl 0738.47018)