## Advances in Operator Theory

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

### Closedness and invertibility for the sum of two closed operators

#### 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

**Mathematical Reviews number (MathSciNet)**

MR3795101

**Zentralblatt MATH identifier**

06902453

**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