Nonlinear Hilbert adjoints: properties and applications to Hankel singular value analysis
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.
scientific article; zbMATH DE number 1836620
scientific article