Zur Eigenwerttheorie einer Klasse symmetrischer Operatoren

Heuser, Harro

doi:10.1007/bf01180481

Cited by 4 publications

(15 citation statements)

References 3 publications

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“…The results in this section represent an extension to the eigenvalue problem (2-1) of the corresponding results for the selfadjoint differential and abstract eigenvalue problems obtained in [5,8,10,12,14,18,25,36]. At the same time we also obtain some additional new results.…”

Section: T H E E Ig E N V a L U E Pr O B Le M T U -Xsu =mentioning

confidence: 55%

“…The relation (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) shows that F{u) attains its minimum at This shows that e Dr and therefore, by lemma 2-2, wx is an ordinary eigenelement belonging to eigenvalue A: of (2-1).…”

Section: \ Q(wlimentioning

confidence: 96%

“…To prove theorem 2-1 ( b )i t is sufficient, in view of lemma 2-2, to show that To do this let us first observe that since, by theorem 2-1 (a), is a generalized eigenelement we have for all u in H0a nd in particular for all in the identity [wx, u\ = A^ztq, ( ), (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) othesis on Sa nd lemma 2-3, can be written in the form…”

Section: \ Q(wlimentioning

confidence: 99%

“…[ Published 11 January 1968 However, the investigation of both of these problems for (i) was limited mostly to selfadjoint bounded and unbounded differential and abstract operators. Thus, when T and S are selfadjoint operators, the existence problem was considered in [4,5,8,10,14,25] in case the operators are positive definite and in [12,18,36] in case is the identity and the spectrum of ^contains at most eigenvalues of finite multiplicity. The investigation of the ap proximation problem, both for bounded and unbounded operators, led to the development of a group of direct methods of Ritz, Galerkin, moments, and others [4,5,10,14,15,16,25,27] and to a group of iterative methods of gradient type [1,2,3,6,11,13,16,17,19,20,22,29,30,33,37].…”

Section: Introductionmentioning

confidence: 99%

“…After stating in § 1 certain properties of A sym m etric and / / 0-bounded operators, in § 2 we consider the problem of existence of eigenvalues and eigenvectors of (i) and their pro perties under various conditions on T and S .Accordingly problem (i) the corresponding results in [8,25] while in § 2-3 we consider conditions under which the results derived in [14,36,18,12] are also valid for (i). Using the properties of / / 0-bounded operators we prove stronger assertions concerning the Fourier expansion and the completeness of eigenvectors than those obtained in [18,12].…”

Section: Introductionmentioning

confidence: 99%

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On the eigenvalue problem Tu -λ Su =0 with unbounded and nonsymetric operators T and S

1968

Phil. Trans. R. Soc. Lond. A

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In this paper we study both the theoretical problem of the existence and the practical problem of the approximate calculation of eigenvalues and eigenvectors of (i) = 0, where T and S are some linear (in general unbounded and nonhermitian) operators in a Hilbert space. After a short discussion of a class of K -symmetric operators, in section 2 the author proves the existence of eigenvalues and eigenvectors of (i) under various conditions on T and S and investigates conditions under which the set of eigenvectors of (i) is complete. Section 3 indicates briefly the applicability and the unifying property of the generalized method of moments to the approximate solution of (i). Section 4 presents and thoroughly studies a very general new iterative method for the approximate solution of (i). The advantage of this method is that it does not require the practically inconvenient preliminary reduction of (i) to an equivalent problem with bounded operators and that under certain rather general conditions the convergence is mono tonic. Furthermore, by specializing operators and parameters, our iterative method contains as a special case, almost every known iterative method for the calculation of eigenvalues (mostly proved previously only for symmetric matrices and bounded operators). Finally the applicability and the numerical effectiveness of the iterative method is illustrated by calculating the smallest eigenvalue for a selfadjoint and nonselfadjoint eigenvalue problems arising in the problems of elastic stability.

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