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The \(H^\infty\)-calculus and sums of closed operators. (English) Zbl 0992.47005

The authors develop a general operator-valued functional calculus for operators with an \(H^{\infty}\)-calculus in a Banach space. The results are applied to the join functional calculus of two sectorial operators when one has an \(H^{\infty}\) calculus. Specifically, sufficient conditions on sectorial operators \(A, B\) for the sum \(A+B\) to be closed are presented. The authors also give applications to the problem of maximal \(L_p\)-regularity of the Cauchy problem \[ y'(t)+Ay(t)=f(t), \quad y(0)=0. \] The presented results emphasize the fact that an \(H^{\infty}\)-calculus induces an unconditional expansion of the identity of the underlying Banach space. With the use of this observation it is shown how classical results on unconditional bases due to Lindenstrauss and Pelczynski can be recast as results on operators with an \(H^{\infty}\)-calculus on \(L_1\) and \(C(K)\)-spaces.

MSC:

47A60 Functional calculus for linear operators
47D06 One-parameter semigroups and linear evolution equations
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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