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\(H^ \infty\) functional calculus for perturbations of generators of holomorphic semigroups. (English) Zbl 0738.47018

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)

MSC:

47A60 Functional calculus for linear operators
46H30 Functional calculus in topological algebras
47D06 One-parameter semigroups and linear evolution equations
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