Operator representation of sectorial linear relations and applications

Gerald Wanjala*

*Corresponding author for this work

Research output: Contribution to journalArticle

Abstract

Let ℋ be a Hilbert space with inner product 〈⋅,⋅〉 and let T be a non-densely defined linear relation in ℋ with domain D(T). We prove that if T is sectorial then it can be expressed in terms of some sectorial operator A with domain D(A)=D(T) and that T is maximal sectorial if and only if A is maximal sectorial in D(T)¯. The operator A has the property that for every u∈D(A) and every v∈D(T) and any u′∈T(u), 〈Au,v〉=〈u′,v〉. We use this representation to show that every sectorial linear relation T is form closable, meaning that the form associated with T has a closed extension. We also prove a result similar to Kato’s first representation theorem for sectorial linear relations. Unlike the results available in the literature, we do not assume that the graph of the linear relation T is a closed subspace of H×H.

Original languageEnglish
Pages (from-to)1-16
Number of pages16
JournalJournal of Inequalities and Applications
Volume2015
Issue number1
DOIs
Publication statusPublished - 2015

Keywords

  • linear form
  • numerical range
  • sectorial linear relation

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Discrete Mathematics and Combinatorics

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