Self-adjoint extensions of symmetric subspaces

Dijksma, Aad; Snoo, H.S.V. de

doi:10.2140/pjm.1974.54.71

Cited by 115 publications

(92 citation statements)

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“…with TcT, then T" is called an extension of Γ. If in this case φ' -φ, then T" is said to be a canonical extension of Γ. Evidently TaT is equivalent to K Zo (T)c (£ Z0 (T') It is well known (see e.g., [4]) that every symmetric c.l.r. admits selfadjoint extensions and that it admits canonical selfadjoint extensions if and only if its defect numbers n + and n_ are equal.…”

Section: On Generalized Resolvents and Q-functions 137mentioning

confidence: 99%

“…It can be shown [4] that this definition is correct, that is the property (2) is independent of the point z o eC+.…”

Section: On Generalized Resolvents and Q-functions 137mentioning

confidence: 99%

“…In some problems related to the spectral theory in Hubert space it is more natural and at the same time often less restrictive to use symmetric linear relations (in the terminology of [1], subspaces in the terminology of [2][3][4]) instead of symmetric operators. Hence the question arises if the theory of generalized resolvents of symmetric operators can be extended to symmetric linear relations.…”

mentioning

confidence: 99%

“…Hence the question arises if the theory of generalized resolvents of symmetric operators can be extended to symmetric linear relations. In [4] a description of all generalized resolvents of a symmetric linear relation was given, following the lines of A. V. Straus [5] in the operator case. It is the aim of this paper to generalize M. G. Kreϊn's formula for the generalized resolvents of a symmetric operator (see [6,7]) to the symmetric linear relation case.…”

mentioning

confidence: 99%

“…A closed linear relation (c.l.r.) in § is a (closed) subspace of φ 2 = & © Φ (see [2], [3], [4]). Evidently, the graph of a closed linear operator in ^ is a c.l.r.…”

mentioning

confidence: 99%

See 4 more Smart Citations

On generalized resolvents andQ-functions of symmetric linear relations (subspaces) in Hilbert space

Langer¹,

Textorius²

1977

Pacific J. Math.

162

167

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Section: On Generalized Resolvents and Q-functions 137mentioning

confidence: 99%

“…It can be shown [4] that this definition is correct, that is the property (2) is independent of the point z o eC+.…”

Section: On Generalized Resolvents and Q-functions 137mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…A closed linear relation (c.l.r.) in § is a (closed) subspace of φ 2 = & © Φ (see [2], [3], [4]). Evidently, the graph of a closed linear operator in ^ is a c.l.r.…”

mentioning

confidence: 99%

See 3 more Smart Citations

On generalized resolvents andQ-functions of symmetric linear relations (subspaces) in Hilbert space

Langer¹,

Textorius²

1977

Pacific J. Math.

162

167

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Some Questions in the Perturbation Theory ofJ-Nonnegative Operators in Krein Spaces

Jonas¹,

Langer²

1983

Math. Nachr.

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In this note we study perturbations of a J-nonnegative operator A in a KREIN space which are such that the difference of the resolvents of A and of the perturbed operator B is of rank one. Here B is also supposed to be J-selfadjoint. With the pair A, B we associate a one-parameter family {Br),,eR of J-selfadjoint operators (or linear relations) which extend the "common part" of A and B. In particular, A and B belong to this family. Furthermore, we characterize those opemtors in the family which are J-nonnegative (Theorem 2.3.2). The description of the operators in this family through the corresponding resolvents is similar to M. G. KREIN'S description of the orthogonal generalized resolvents of a EhsmTian operator with defect numbers one in HILBHRT space (see e.g.[l]). The Theorem 2.3.2 gives, in particular, a description of the generalized resolvents of a densely defined J-nonnegative HrPmmian operator A, with defect one on some KBEIN space S, which are generated by J-nonnegative J-selfadjoint extensions of A, in S. ( c q 1~2 & ( A ) g, t p 2 & ( L l ) g) 5 0-1.Hence the sequence G1'zgc(Ll) 8 converges to aome h E .#' 0 N ( C ) if d tenda to (0, oo), and llhllz 5 c-l. Making use of the second relation in (8) we find (Ci/z)h = fi, and the assertion follows. Proof of Proposition2.1. Sincetheseta9(Aa) andO(H) aredeneein.#'~~lz =.#'+l/2(H) the operator A,, is J-nonnegative if and only if 0 5 [Jz + a[z, { I f , 21 = (J& z) + a I(Jf, 41' (z E Jf'+1,z(H)), and this holds if and only if (9) ( H z , 5) 2 --d~ I(J1, %)I8 (2 E OW)). We have Jf E 2-l/2(H). Then, according to Lemma 2.2, the relation (9) holds if and only if either u 2 0 or Jf E B((Hi")-) and 0 < -u 5 l\((Hi'z)-)-l Jfll-8. Now observe that Hi/2 = JKi12J. Hence Jj = ( H i / 2 )z, where z E 2 0 J ( H ) r ie equivalent to f = (KiI2)-Jz. It follows that ll((Hi/z)-)-l Jfll = ll((Ki/2)-)-1 1 1 1 and the Proposition 2.1 is proved. y > IR(i) 11: if 111-= 00 y > IB(i) fl; or y 5 fl', -lfl! if lfl-< 00. Moreover the relation f = a8(i) f -i8(i) f implies that IfI-< 00 if and only if @(i) fl-< 00. A little computation yields in thie caw the relation By (6) the inequality (6) is equivalent to and to I~I ? = IB(i) fI: + IR(i) 11: . Thus, the Proposition 2.1 has the following Corollary. Zf f E then B, ie J-mntnegatave ij and only if y E R \ (-I &) f t , I&) 11;).Here the nonnegativity of BI;ci,fl: follows from Lemma 1.2.3. The general m e . In order to find the valuea of the parameter y for which B, is J-nonnegative in the general case 1 E #'-we shall use the following proposition. Proposition 3.1. Let A be a J-nonnegative J-eelfadjoint opemtor, e(A) $; 0 and put H := J A . Then for arbitrary x E .#' the function (0, 00) 3 7 -.ra[AR(iq) 2, R(irl) 4 ia nondecreasing. Z j z E 94(H1I2) then lim qz[AR(iq) z, R(iq) z] = ( H l k , EI1l2z), st-

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Selbstadjungierte Relations-Erweiterungen für eine Klasse von symmetrischen Differential-Funktional-Operatoren

Timmel

1985

Math. Nachr.

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show abstract

Self-adjoint extensions of symmetric subspaces

Cited by 115 publications

References 8 publications

On generalized resolvents andQ-functions of symmetric linear relations (subspaces) in Hilbert space

On generalized resolvents andQ-functions of symmetric linear relations (subspaces) in Hilbert space

Some Questions in the Perturbation Theory ofJ-Nonnegative Operators in Krein Spaces

Selbstadjungierte Relations-Erweiterungen für eine Klasse von symmetrischen Differential-Funktional-Operatoren

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