Advances in Operator Theory

Closedness and invertibility for the sum of two closed operators

Nikolaos Roidos

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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.

Article information

Source
Adv. Oper. Theory Volume 3, Number 3 (2018), 582-605.

Dates
Received: 19 January 2018
Accepted: 8 February 2018
First available in Project Euclid: 3 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.aot/1520046046

Digital Object Identifier
doi:10.15352/aot.1801-1297

Subjects
Primary: 47A60: Functional calculus
Secondary: 47A05: General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47A10: Spectrum, resolvent 35K90: Abstract parabolic equations

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

Citation

Roidos, Nikolaos. Closedness and invertibility for the sum of two closed operators. Adv. Oper. Theory 3 (2018), no. 3, 582--605. doi:10.15352/aot.1801-1297. https://projecteuclid.org/euclid.aot/1520046046


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