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Summer 2018 Closedness and invertibility for the sum of two closed operators
Nikolaos Roidos
Adv. Oper. Theory 3(3): 582-605 (Summer 2018). DOI: 10.15352/aot.1801-1297

Abstract

We show a Kalton–Weis type theorem for the general case of noncommuting operators. More precisely, we consider sums of two possibly noncommuting linear operators defined in a Banach space such that one of the operators admits a bounded $H^\infty$-calculus, the resolvent of the other one satisfies some weaker boundedness condition and the commutator of their resolvents has certain decay behavior with respect to the spectral parameters. Under this consideration, we show that the sum is closed and that after a sufficiently large positive shift it becomes invertible and moreover sectorial. As an application we recover a classical result on the existence, uniqueness, and maximal $L^{p}$-regularity for solutions of the abstract linear nonautonomous parabolic problem.

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Nikolaos Roidos. "Closedness and invertibility for the sum of two closed operators." Adv. Oper. Theory 3 (3) 582 - 605, Summer 2018. https://doi.org/10.15352/aot.1801-1297

Information

Received: 19 January 2018; Accepted: 8 February 2018; Published: Summer 2018
First available in Project Euclid: 3 March 2018

zbMATH: 06902453
MathSciNet: MR3795101
Digital Object Identifier: 10.15352/aot.1801-1297

Subjects:
Primary: 47A60
Secondary: 35K90 , 47A05 , 47A10

Keywords: abstract Cauchy problem , bounded $H^{\infty}$-calculus , maximal regularity , sectorial operators

Rights: Copyright © 2018 Tusi Mathematical Research Group

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Vol.3 • No. 3 • Summer 2018
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