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On the inverse of the sum of two sectorial operators

Nikolaos Roidos

Abstract

We study an abstract linear operator equation on a Banach space by using the inverse of the sum of two sectorial operators. We prove that the boundedness of a special type of operator valued $H^\infty$-calculus is sufficient for maximal regularity of the solution. We apply the result to the abstract parabolic problem, to give a maximal $L^{p}$-regularity condition. We also study the abstract hyperbolic problem and give a sufficient condition for the existence of solution.

On the inverse of the sum of two sectorial operators

Abstract

We study an abstract linear operator equation on a Banach space by using the inverse of the sum of two sectorial operators. We prove that the boundedness of a special type of operator valued -calculus is sufficient for maximal regularity of the solution. We apply the result to the abstract parabolic problem, to give a maximal -regularity condition. We also study the abstract hyperbolic problem and give a sufficient condition for the existence of solution.

Paper Structure

This paper contains 5 sections, 3 theorems, 61 equations.

Table of Contents

  1. introduction
  2. The inverse of A+B
  3. The abstract parabolic problem
  4. The abstract hyperbolic problem
  5. Appendix

Key Result

Theorem 2.1

Let $E$ be a Banach space, $A\in\mathcal{P}(\theta_{A})$ and $B\in\mathcal{P}(\theta_{B})$ be resolvent commuting with $\theta_{A}>\theta_{B}$ and $\theta_{A}+\theta_{B}>\pi$. Then, $A+B$ with $\mathcal{D}(A+B)=\mathcal{D}(A)\cap\mathcal{D}(B)$ is closable and the following equation for any $y\in E$, has a unique solution $x\in\bigcap_{\theta<1}(E,\mathcal{D}(A))_{\theta,q}\cap(E,\mathcal{D}(B))_

Theorems & Definitions (11)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • Definition 4.1
  • Theorem 4.1
  • proof
  • ...and 1 more