Nonlinear Hilbert adjoints: properties and applications to Hankel singular value analysis
type: Item
instance of: scholarly article
title: Nonlinear Hilbert adjoints: properties and applications to Hankel singular value analysis
zbMATH Open document ID: 1025.47043
DOI: 10.1016/S0362-546X(01)00867-7
author: Jacquelien M. A. Scherpen
author: W. Steven Gray
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-11-28T00:00:00Z
review text: This article deals with the notion of adjoint operator for nonlinear operators acting between two Hilbert spaces \(E\) and \(F\). If an operator \({\mathcal F}\) acts between \(E\) and \(F\), its adjoint operator \({\mathcal F}^*: F\times E\to E\) is defined by the equation \[ \langle{\mathcal F}, y\rangle_F= \langle u,{\mathcal F}^*(y, u)\rangle_E; \] of course, \({\mathcal F}^*(y, u)\) is linear in \(y\). In the case when the domain of \({\mathcal F}\) is any convex neighborhood of \(0\), \({\mathcal F}0= 0\), and \({\mathcal F}\) is continuously differentiable, the mapping \[ {\mathcal F}^*(y, u)= \int^1_0({\mathcal F}'(tu))^*(y) dt \] is an adjoint of \({\mathcal F}\) on \(F\times U\). Some properties of these adjoints are proved in the article. In particular, the identity \(({\mathcal G}{\mathcal F})^*= {\mathcal F}^*({\mathcal G}^*{\mathcal F})\), formulas for the second adjoint, some analogues for the classical \(C^*\)-identity in the theory of \(C^*\)-algebras, three equations relating adjoints of a nonlinear operator and usual adjoints of its derivatives, the theorem about the connection between of the spectrum of a continuously differentiable nonlinear operator and the spectrum of its derivatives, and so on. It should be noted that there is a number of interesting examples of nonlinear operators and their adjoints in the article; in particular, Hankel nonlinear operators are considered in detail.
Mathematics Subject Classification ID: 47J10
Mathematics Subject Classification ID: 46T20
Mathematics Subject Classification ID: 47H30
zbMATH DE Number: 1836620
zbMATH Keywords: adjoint of derivatives
zbMATH Keywords: nonlinear Hankel operators
zbMATH Keywords: nonlinear operators
zbMATH Keywords: Hilbert spaces
zbMATH Keywords: adjoint operator
zbMATH Keywords: second adjoint
zbMATH Keywords: spectrum
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
cites work: Manifolds, tensor analysis, and applications.
cites work: Q4940796
cites work: Some spectral theory for nonlinear operators
cites work: Nonlinear compact mappings and their adjoints
cites work: Q5655597
cites work: Nonlinear eigenvalue problems and Galerkin approximations
cites work: Some properties of nonlinear adjoint operators
cites work: Duals and propagators: A canonical formalism for nonlinear equations
cites work: Q3971314
cites work: Variational and Hamiltonian control systems
cites work: An algebraic approach to nonlinear functional expansions
cites work: Eigenvectors for nonlinear maps
cites work: Eigenstructure of nonlinear hankel operators
cites work: Nonlinear input-normal realizations based on the differential eigenstructure of Hankel operators
cites work: Hamiltonian realizations of nonlinear adjoint operators
cites work: Contributions to the spectral theory for nonlinear operators in Banach spaces
cites work: A nonlinear spectral approach to surjectivity in Banach spaces
cites work: On the nonuniqueness of singular value functions and balanced nonlinear realizations
cites work: Hankel singular value functions from Schmidt pairs for nonlinear input-output systems
cites work: Nonlinear Eigenproblems
cites work: Eigenvalues and eigenvectors of nonlinear operators and applications
cites work: Singular value analysis of deformable systems
cites work: Q5526485
cites work: Q4023346
cites work: Principal component analysis in linear systems: Controllability, observability, and model reduction
cites work: Q5645931
cites work: Existence of a spectrum for nonlinear transformations
cites work: On differentiability and surjectivity of \(\alpha\)-Lipschitz mappings
cites work: Q5664372
cites work: Balancing for nonlinear systems
cites work: Minimality and local state decompositions of a nonlinear state space realization using energy functions
cites work: Normalized coprime factorizations and balancing for unstable nonlinear systems
cites work: Computation of a functional in a certain nonlinear problem using the adjoint equation
cites work: Functional analysis: spectral theory
cites work: The adjoints of differentiable mappings
cites work: The closure of the numerical range contains the spectrum
cites work: Secondary operational calculus
cites work: Q3128879
full work available at URL: s0362-546x(01)00867-7
OpenAlex ID: W2095835057
description: scientific article; zbMATH DE number 1836620