Operator representation of sectorial linear relations and applications

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• 作者：Gerald Wanjala
• 关键词：47A06 ; 47A07 ; 47A12 ; 47B44 ; linear form ; sectorial linear relation ; numerical range
• 刊名：Journal of Inequalities and Applications
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：2015
• 期：1
• 全文大小：1,288 KB
• 参考文献：1. Arens, R: Operational calculus of linear relations. Pac. J. Math. 11, 9-23 (1961) CrossRef
  2. Coddington, EA, de Snoo, HSV: Positive selfadjoint extensions of positive symmetric subspaces. Math. Z. 159, 203-214 (1978) CrossRef
  3. Coddington, EA: Extension theory of formally normal and symmetric subspaces. Mem. Am. Math. Soc. 134, 1-80 (1973)
  4. Kato, T: Perturbation Theory for Linear Operators. Springer, New York (1980)
  5. Cross, R: Multivalued Linear Operators. Dekker, New York (1998)
  6. Arlinskii, YM: Boundary conditions for maximal sectorial extensions of sectorial operators. Math. Nachr. 209, 5-36 (2000) CrossRef
  7. Arlinskii, YM, Popov, AB: On sectorial matrices. Linear Algebra Appl. 370, 133-146 (2003) CrossRef
  8. Malamud, M: Operator holes and extensions of sectorial operators and dual pairs of contractions. Math. Nachr. 279, 625-655 (2006) CrossRef
  
• 刊物主题：Analysis; Applications of Mathematics; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1029-242X

文摘

Let ?be a Hilbert space with inner product \(\langle\cdot, \cdot\rangle\) and let \(\mathcal{T}\) be a non-densely defined linear relation in ?with domain \(D(\mathcal{T})\) . We prove that if \(\mathcal{T}\) is sectorial then it can be expressed in terms of some sectorial operator A with domain \(D(A) = D(\mathcal{T})\) and that \(\mathcal{T}\) is maximal sectorial if and only if A is maximal sectorial in \(\overline{D(\mathcal{T})}\) . The operator A has the property that for every \(u\in D(A)\) and every \(v\in D(\mathcal{T})\) and any \(u^{\prime}\in\mathcal{T}(u)\) , \(\langle Au,v\rangle= \langle u^{\prime}, v\rangle\) . We use this representation to show that every sectorial linear relation \(\mathcal{T}\) is form closable, meaning that the form associated with \(\mathcal{T}\) 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 \(\mathcal{T}\) is a closed subspace of \(\mathcal{H} \times\mathcal{H}\) .