Variational Methods for One and Several Parameter Non-Linear Eigenvalue Problems

1981 ◽  
Vol 33 (1) ◽  
pp. 210-228 ◽  
Author(s):  
Paul Binding
Keyword(s):  
Hilbert Space ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Linear Operators ◽  
Main Thrust ◽  
Non Linear ◽  
Multiparameter Eigenvalue Problem ◽  
The One ◽  
Image Position ◽  
Parameter Problem

We shall consider a multiparameter eigenvalue problem of the form(1.1)where λ ∈ Rk while Tn and Vn(λ) are self-adjoint linear operators on a Hilbert space Hn. If λ = (λ1, …, λk) ∈ Rk and satisfy (1.1) then we call λ an eigenvalue, x an eigenvector and (λ, x) an eigenpair. While our main thrust is towTards the general case of several parameters λn, the method ultimately involves reduction to a sequence of one parameter problems. Our chief contributions are (i) to generalise the conditions under which this reduction is possible, and (ii) to develop methods for the one parameter problem particularly suited to the multiparameter application. For example, we give rather general results on the magnitude and direction of the movement of non-linear eigenvalues under perturbation.

Existence conditions for eigenvalue problems generated by compact multiparameter operators

1984 ◽  
Vol 96 (3-4) ◽  
pp. 261-274 ◽  
Author(s):  
Paul Binding ◽  
Patrick J. Browne ◽  
Lawrence Turyn
Keyword(s):  
Hilbert Space ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Compact Resolvent ◽  
Multiparameter Eigenvalue Problem ◽  
Existence Conditions ◽  
Image Position ◽  
Adjoint Operators

SynopsisLet T, V1,…, Vk denote compact symmetric linear operators on a separable Hilbert space H, and write W(λ) = T + λ1V1 + … + λkVk, λ = (λ1, …, λk) ϵ ℝk. We study conditions on the conerelated to solubility of the multiparameter eigenvalue problemwith W(λ)−I nonpositive definite. The main result is as follows.Theorem. If 0 ∉ V, then (*) is soluble for any T. If 0 ∈ V, then there exists T such that (*) is insoluble.We also deduce analogous results for problems involving self-adjoint operators with compact resolvent.

scholarly journals Asymptotic spectrum of multiparameter eigenvalue problems

1996 ◽  
Vol 39 (1) ◽  
pp. 119-132 ◽  
Author(s):  
Hans Volkmer
Keyword(s):  
Hilbert Space ◽  
Maximum Principle ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Asymptotic Spectrum ◽  
Multiparameter Eigenvalue Problem ◽  
Sturm Liouville

Results are given for the asymptotic spectrum of a multiparameter eigenvalue problem in Hilbert space. They are based on estimates for eigenvalues derived from the minim un-maximum principle. As an application, a multiparameter Sturm-Liouville problem is considered.

VI.—On Solvability of Some Two-Parameter Eigenvalue Problems in Hilbert Space

1968 ◽  
Vol 68 (1) ◽  
pp. 83-93
Author(s):  
Z. Bohte
Keyword(s):  
Hilbert Space ◽  
Power Series ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Radius Of Convergence ◽  
Finite Radius ◽  
Image Position ◽  
Adjoint Operators ◽  
Two Parameter

SynopsisThis paper studies two particular cases of the general 2-parameter eigenvalue problem namelywhere A, B, B1, B2, C, C1, C2 are self-adjoint operators in Hilbert space, all except A being bounded. The disposable parameters λ and μ have to be determined so that the equations have non-trivial solutions x, y.On the assumption that the solution is known for ∊ = o, solutions are constructed in the form of series for λ, μ, x, y as power series in ∊ with finite radius of convergence.

scholarly journals Multiparameter spectral theory and Taylor's joint spectrum in Hilbert space

1988 ◽  
Vol 31 (1) ◽  
pp. 127-144 ◽  
Author(s):  
B. P. Rynne
Keyword(s):  
Hilbert Space ◽  
Spectral Theory ◽  
Linear Operators ◽  
Coupled Systems ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Joint Spectrum ◽  
Image Position

Let n≧1 be an integer and suppose that for each i= 1,…,n, we have a Hilbert space Hi and a set of bounded linear operators Ti, Vij:Hi→Hi, j=1,…,n. We define the system of operatorswhere λ=(λ1,…,λn)∈ℂn. Coupled systems of the form (1.1) are called multiparameter systems and the spectral theory of such systems has been studied in many recent papers. Most of the literature on multiparameter theory deals with the case where the operators Ti and Vij are self-adjoint (see [14]). The non self-adjoint case, which has received relatively little attention, is discussed in [12] and [13].

scholarly journals Factorization of hyponormal operators

1972 ◽  
Vol 13 (3) ◽  
pp. 323-326
Author(s):  
Mary R. Embry
Keyword(s):  
Hilbert Space ◽  
Linear Operators ◽  
Continuous Linear ◽  
Hyponormal Operators ◽  
Image Position ◽  
Continuous Linear Operators

In [1]R. G. Douglas proved that if A and B are continuous linear operators on a Hilbert space X, the following three conditions are equivalent:

scholarly journals Positive functionals and oscillation criteria for second order differential systems

1979 ◽  
Vol 22 (3) ◽  
pp. 277-290 ◽  
Author(s):  
Garret J. Etgen ◽  
Roger T. Lewis
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Second Order ◽  
Linear Operators ◽  
The Self ◽  
Differential Systems ◽  
Bounded Linear ◽  
Positive Functionals ◽  
Image Position ◽  
Adjoint Operators

Let ℋ be a Hilbert space, let ℬ = (ℋ, ℋ) be the B*-algebra of bounded linear operators from ℋ to ℋ with the uniform operator topology, and let ℒ be the subset of ℬ consisting of the self-adjoint operators. This article is concerned with the second order self-adjoint differential equation

scholarly journals Application of the Ritz method to non-standard eigenvalue problems

1969 ◽  
Vol 10 (3-4) ◽  
pp. 367-384 ◽  
Author(s):  
A. L. Andrew
Keyword(s):  
Eigenvalue Problems ◽  
Ritz Method ◽  
Positive Definite ◽  
Linear Operators ◽  
Extensive Literature ◽  
Maximum Eigenvalue ◽  
Special Cases ◽  
Image Position ◽  
Linear Manner

There is an extensive literature on application of the Ritz method to eigenvalue problems of the type where L1, L2 are positive definite linear operators in a Hilbert space (see for example [1]). The classical theory concerns the case in which there exists a minimum (or maximum) eigenvalue, and subsequent eigenvalues can be located by a well-known mini-max principle [2; p. 405]. This paper considers the possibility of application of the Ritz method to eigenvalue problems of the type (1) where the linear operators L1L2 are not necessarily positive definite and a minimum (or maximum) eigenvalue may not exist. The special cases considered may be written with the eigenvalue occurring in a non-linear manner.

scholarly journals States which have a trace-like property relative to a C*-subalgebra of B(H)

1976 ◽  
Vol 17 (2) ◽  
pp. 158-160
Author(s):  
Guyan Robertson
Keyword(s):  
Hilbert Space ◽  
Spectral Radius ◽  
Operator Theory ◽  
Linear Operators ◽  
Operator Norm ◽  
Identity Operator ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Image Position

In what follows, B(H) will denote the C*-algebra of all bounded linear operators on a Hilbert space H. Suppose we are given a C*-subalgebra A of B(H), which we shall suppose contains the identity operator 1. We are concerned with the existence of states f of B(H) which satisfy the following trace-like relation relative to A:Our first result shows the existence of states f satisfying (*), when A is the C*-algebra C*(x) generated by a normaloid operator × and the identity. This allows us to give simple proofs of some well-known results in operator theory. Recall that an operator × is normaloid if its operator norm equals its spectral radius.

scholarly journals An algorithm for solving generalized algebraic Lyapunov equations in Hilbert space, applications to boundary value problems

1988 ◽  
Vol 31 (1) ◽  
pp. 99-105 ◽  
Author(s):  
Lucas Jódar
Keyword(s):  
Hilbert Space ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Operator Equation ◽  
Boundary Value ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Space Applications ◽  
Bounded Linear ◽  
Image Position

Let L(H) be the algebra of all bounded linear operators on a separable complex Hubert space H. In a recent paper [7], explicit expressions for solutions of a boundary value problem in the Hubert space H, of the typeare given in terms of solutions of an algebraic operator equation

Spectral Inclusion and C.N.E.

1982 ◽  
Vol 34 (4) ◽  
pp. 883-887 ◽  
Author(s):  
A. R. Lubin
Keyword(s):  
Hilbert Space ◽  
Linear Operators ◽  
Normal Extension ◽  
Unitary Equivalence ◽  
Bounded Linear ◽  
Multi Index ◽  
Reducing Subspace ◽  
Normal Operators ◽  
Image Position ◽  
Minimal Normal Extension

1. An n-tuple S = (S1, …, Sn) of commuting bounded linear operators on a Hilbert space H is said to have commuting normal extension if and only if there exists an n-tuple N = (N1, …, Nn) of commuting normal operators on some larger Hilbert space K ⊃ H with the restrictions Ni|H = Si, i = 1, …, n. If we takethe minimal reducing subspace of N containing H, then N is unique up to unitary equivalence and is called the c.n.e. of S. (Here J denotes the multi-index (j1, …, jn) of nonnegative integers and N*J = N1*jl … Nn*jn and we emphasize that c.n.e. denotes minimal commuting normal extension.) If n = 1, then S1 = S is called subnormal and N1 = N its minimal normal extension (m.n.e.).

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