2001
DOI: 10.1006/jmaa.2001.7703
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On Krein's Formula in the Case of Non-densely Defined Symmetric Operators

Abstract: In this paper we provide some additional results related to Krein's resolvent formula for a non-densely defined symmetric operator. We show that coefficients in Krein's formula can be expressed in terms of analogues of the classical von Neumann formulas. The relationship between two Weyl-Tichmarsh m-functions corresponding to self-adjoint extensions of a non-densely defined symmetric operator is established.  2001 Elsevier Science

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Cited by 8 publications
(8 citation statements)
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“…The estimates in (C.11) follow from combining (C.1), (C.9), and (C.10). The estimates in (C.12) follow from the fact that ∂ j E n (z; x) = −2πx j E n+2 (z; x), z ∈ C\{0}, x ∈ R n \{0}, 1 j n, n 2, (C. 14) which permits one to reduce them essentially to (C.11) with n replaced by n + 2. The recursion relation (C.14) is a consequence of the well-known identity (cf.…”
Section: Appendix C Estimates For the Fundamental Solution Of The Hementioning
confidence: 99%
“…The estimates in (C.11) follow from combining (C.1), (C.9), and (C.10). The estimates in (C.12) follow from the fact that ∂ j E n (z; x) = −2πx j E n+2 (z; x), z ∈ C\{0}, x ∈ R n \{0}, 1 j n, n 2, (C. 14) which permits one to reduce them essentially to (C.11) with n replaced by n + 2. The recursion relation (C.14) is a consequence of the well-known identity (cf.…”
Section: Appendix C Estimates For the Fundamental Solution Of The Hementioning
confidence: 99%
“…25) is a self-adjoint operator with the property that − ∆ min − zI Ω ⊆ S ⊆ −∆ max − zI Ω , (1. 26) then there exist X, a closed subspace of N 1/2 (∂Ω) * , and L : dom(L) ⊆ X → X * , a self-adjoint operator, such that S = −∆ D X,L,z .…”
Section: Introductionmentioning
confidence: 99%
“…Actually, it was Naȋmark [605,606,607] who investigated different types of extensions (with exit) and a proper interpretation then yields the corresponding resolvent formula. For the notion of Štraus family and for Kreȋn's formula for exit space extensions we refer to [491,492,679] and to [726,727,728,729,730,731,732,733]; some other related papers are [51,121,126,219,234,237,254,266,320,497,523,554,595,596,624]. The Štraus families are restrictions of the adjoint in terms of "λ-dependent boundary conditions" given by a Nevanlinna family or corresponding Nevanlinna pair.…”
Section: Notes On Chaptermentioning
confidence: 99%