Spectral theory and functional calculus for operators on spaces of generalized functions
type: Item
instance of: scholarly article
title: Spectral theory and functional calculus for operators on spaces of generalized functions
zbMATH Open document ID: 0783.47016
DOI: 10.1016/0022-247X(92)90291-K
author: S. Singh
published in: Journal of Mathematical Analysis and Applications
publication date: 1992-09-27T00:00:00Z
review text: Suppose \(T\) is a densely defined operator on a complex separable Hilbert space \(H\). Then \(D_ \infty= \bigcap_{n=1}^ \infty D(T^ n)\) becomes a Fréchet space in the topology determined by the seminorms \(\| T^ n x\|\). This space is dense in \(H\) and its dual \(D_ \infty'\) is sequentially complete. The extension \(\overline {T}\) of \(T\) on \(D_ \infty'\) is defined by the relation \(\langle \overline{T}y,x\rangle= \langle y,Tx\rangle\), \(\forall x\in D_ \infty\), \(y\in D_ \infty'\) i.e. this is the \(D_ \infty\) adjoint of \(T\). Now let \(E(\lambda)\) be the spectral function of \(T\). For every Borel measurable function \(F\) on \(\mathbb{R}\), the operator \(F(T)\) is defined by \(F(T)x=\int_ R F(\lambda)dE(\lambda)x\). The main theorem in the paper describes all such functions which give rise to continuous operators on \(D_ \infty\). Namely, \(F(T)\) is continuous on \(D_ \infty\), iff \(\sup_{\lambda\in\sigma(T)} | F(\lambda)| (1+\lambda^ 2)^{- m}= M<\infty\), for some \(m>0\). By definition, \(\overline {F(T)}= F(\overline{T})\). The authors show that \(F\to \overline{F(T)}\in L(D_ \infty)\) is a continuous functional calculus. They also provide an interesting application: starting with \(F_ t(\lambda)=e^{t\lambda}\), \(t>0\), they obtain a semigroup of operators on \(D_ \infty\) which is the solution of a weak abstract Cauchy problem.
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 46F05
zbMATH DE Number: 62879
zbMATH Keywords: densely defined operator on a complex separable Hilbert space
zbMATH Keywords: spectral function
zbMATH Keywords: continuous functional calculus
zbMATH Keywords: weak abstract Cauchy problem
reviewed by: Khristo N. Boyadzhiev
MaRDI profile type: Publication
cites work: Applications of analysis on nilpotent groups to partial differential equations
cites work: Q4126886
cites work: Q3244654
cites work: On Zemanian's distributional eigenfunction transforms
cites work: On the solution of certain distributional boundary value problems
cites work: Orthogonal series representations for generalized functions
cites work: Generalization of Zemanian Spaces of Generalized Functions which Have Orthonormal Series Expansions
cites work: Q4066072
cites work: Q5620934
cites work: Q5571178
full work available at URL: 0022-247x(92)90291-k
OpenAlex ID: W2079680080
description: scientific article; zbMATH DE number 62879