The expansion of a semigroup and a Riesz basis criterion
type: Item
instance of: scholarly article
title: The expansion of a semigroup and a Riesz basis criterion
zbMATH Open document ID: 1131.47042
author: S. P. Yung
author: Gen Qi Xu
published in: Journal of Differential Equations
publication date: 2005-02-22T00:00:00Z
review text: Let \(\mathcal{A}\) be the generator for a strongly continuous semigroup \(T(t)\) on a Hilbert space \(X\). Suppose that the singular set for \(\mathcal{A}\) can be split into two parts \(\sigma(\mathcal{A})=\sigma_1(\mathcal{A})\cup\sigma_2(\mathcal{A})\), where \(\sigma_2(\mathcal{A})\) consists all isolated eigenvalues. Under the assumption that the isolated eigenvalues are contained in a vertical strip, that the multiplicity of the eigenvalues are uniformly bounded and that they satisfy \(\inf_{k\neq l}| \lambda_k-\lambda_l| >0\), the authors prove their main theorem, which says that there exist two \(T(t)\)-invariant subspaces \(X_1\), \(X_2\) such that \(\sigma(\left.\mathcal{A}\right| _{X_1})=\sigma_1(\mathcal{A})\), \(\sigma(\left.\mathcal{A}\right| _{X_2})=\sigma_2(\mathcal{A})\) and \(X_1\oplus X_2\subset X\) (the topological direct sum). In addition, if the Riesz projectors for the eigenvalues satisfy a certain inequality, then \(X=X_1\oplus X_2\). As an application, the authors consider a heat exchanger system with boundary feedback. It is shown that in a certain Hilbert space, the corresponding semigroup fulfils the conditions of the main theorem.
reviewed by: Stefan Jakobsson
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 35F15
Mathematics Subject Classification ID: 47A70
zbMATH DE Number: 2136777
zbMATH Keywords: semigroup expansion
zbMATH Keywords: Riesz basis
zbMATH Keywords: heat exchanger equation
MaRDI profile type: Publication
full work available at URL: j.jde.2004.09.015
OpenAlex ID: W2106418667
cites work: Q4422007
cites work: Q4856165
cites work: Riesz bases formed by exponentials and divided differences
cites work: Q4270280
cites work: Exponential bases in Sobolev spaces in control and observation problems for the wave equation
cites work: Q4329674
cites work: Simultaneous controllability in sharp time for two elastic strings
cites work: On the Stabilization of a Flexible Beam with a Tip Mass
cites work: Q4848744
cites work: Q5654646
cites work: Classes of linear operators. Vol. I
cites work: Interpolation zwischen den Klassen 𝔖p von Operatoren in Hilberträumen
cites work: On the boundary control of a hybrid system with variable coefficients
cites work: Riesz Basis Property and Exponential Stability of Controlled Euler--Bernoulli Beam Equations with Variable Coefficients
cites work: An integrability condition on the resolvent for hyperbolicity of the semigroup
cites work: Stability analysis of heat-exchanger equations with boundary feedbacks
cites work: A splitting problem for unconditional bases of complex exponentials
cites work: Weighted Paley-Wiener spaces
cites work: Q5624013
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Boundary problems for ordinary differential equations with parameter in the boundary conditions
cites work: The Riesz basis property of the system of root vectors for the equation of a nonhomogeneous damped string: Transformation operator method
cites work: Asymptotic and Spectral Analysis of the Spatially Nonhomogeneous Timoshenko Beam Model
cites work: On Buckling Problems
cites work: A functional model approach to linear neutral functional differential equations
cites work: Q4333116
cites work: The Riesz basis property of a Timoshenko beam with boundary feedback and application
cites work: Q3955832
DOI: 10.1016/J.JDE.2004.09.015
description: scientific article; zbMATH DE number 2136777