scholarly journals Global Persistence of the Unit Eigenvectors of Perturbed Eigenvalue Problems in Hilbert Spaces

2020 ◽  
Vol 39 (4) ◽  
pp. 475-497 ◽  
Author(s):  
Pierluigi Benevieri ◽  
Alessandro Calamai ◽  
Massimo Furi ◽  
Maria Patrizia Pera
Keyword(s):  
Hilbert Spaces ◽  
Eigenvalue Problems

Existence conditions for two-parameter eigenvalue problems

1981 ◽  
Vol 91 (1-2) ◽  
pp. 15-30 ◽  
Author(s):  
Paul Binding ◽  
Patrick J. Browne ◽  
Lawrence Turyn
Keyword(s):  
Hilbert Spaces ◽  
Eigenvalue Problems ◽  
Sufficient Conditions ◽  
Compact Resolvent ◽  
Existence Conditions ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Adjoint Operators ◽  
Bounded Below ◽  
Two Parameter

SynopsisWe discuss necessary and sufficient conditions for the existence of eigentuples λ=(λl,λ2) and eigenvectors x1≠0, x2≠0 for the problem Wr(λ)xr = 0, Wr(λ)≧0, (*), where Wr(λ)= Tr + λ1Vr2, r=1,2. Here Tr and Vrs are self-adjoint operators on separable Hilbert spaces Hr. We assume the Vrs to be bounded and the Tr bounded below with compact resolvent. Most of our conditions involve the conesWe obtain results under various conditions on the Tr, but the following is typical:THEOREM. If (*) has a solution for all choices ofT1, T2then (a)0∉ V1UV2,(b)V1∩(—V2) =∅ and (c) V1⊂V2∪{0}, V2⊈V1∪{0}. Conversely, if (a) and (b) hold andV1⊈V2∪∩{0}, V2⊈ then (*) has a solution for all choices ofT1, T2.

scholarly journals Multiparameter root vectors

1989 ◽  
Vol 32 (1) ◽  
pp. 19-29 ◽  
Author(s):  
Paul Binding
Keyword(s):  
Hilbert Spaces ◽  
Eigenvalue Problems ◽  
Finite Codimension ◽  
Image Position ◽  
Recursive Equations

The concept of “root vectors” is investigated for a class of multiparameter eigenvalue problemswhere operate in Hilbert spaces Hm and . Previous work on this “uniformly elliptic” class has demonstrated completeness of the decomposable tensors x1 ⊗…⊗ xk in a subspace G of finite codimension in H=H1 ⊗…⊗ Hk, but questions remain about extending this to a basis of H. In this work, bases of elements ym, in general nondecomposable but satisfying recursive equations of the type are constructed for the “root subspaces” corresponding to λ∈ℝk.

Verified Computations of Eigenvalue Exclosures for Eigenvalue Problems in Hilbert Spaces

2014 ◽  
Vol 52 (2) ◽  
pp. 975-992 ◽  
Author(s):  
Yoshitaka Watanabe ◽  
Kaori Nagatou ◽  
Michael Plum ◽  
Mitsuhiro T. Nakao
Keyword(s):  
Hilbert Spaces ◽  
Eigenvalue Problems ◽  
Verified Computations

scholarly journals Elliptic multiparameter eigenvalue problems

1987 ◽  
Vol 30 (2) ◽  
pp. 215-228 ◽  
Author(s):  
P. A. Binding ◽  
K. Seddighi
Keyword(s):  
Hilbert Spaces ◽  
Eigenvalue Problems ◽  
Positive Definite ◽  
Ellipticity Condition ◽  
Image Position ◽  
Adjoint Operators ◽  
Bounded Below ◽  
Compact Resolvents

We study the eigenproblemwhereand Tm, Vmn are self-adjoint operators on separable Hilbert spaces Hm. We assume the Tm to be bounded below with compact resolvents, and the Vmn to be bounded and to satisfy an “ellipticity” condition. If k = 1 then ellipticity is automatic, and if each Tm is positive definite then the problem is “left definite”.

Self-adjoint subspace extensions satisfying λ-linear boundary conditions

1981 ◽  
Vol 90 (1-2) ◽  
pp. 107-124 ◽  
Author(s):  
Harald Röh
Keyword(s):  
Hilbert Space ◽  
Boundary Conditions ◽  
Hilbert Spaces ◽  
Eigenvalue Problems ◽  
Linear Boundary ◽  
Deficiency Indices ◽  
Symmetric Subspace

SynopsisLet S be a symmetric subspace in a Hilbert space ℋ2 with finite equal deficiency indices and let S* be its adjoint subspace in ℋ2. We consider those self-adjoint subspace extensions ℋ of S into some larger Hilbert spaces ℋ2 = (ℋ × ℂm)2 which satisfy H⋂({0} × ℂm)2 = {{0,0}}. These extensions H are characterized in terms of inhomogeneous boundary conditions for S*; they are associated with eigenvalue problems for S* depending on λ-linear boundary conditions, which we also characterize.

scholarly journals Global Persistence of the Unit Eigenvectors of Perturbed Eigenvalue Problems in Hilbert Spaces: The Odd Multiplicity Case

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 561
Author(s):  
Pierluigi Benevieri ◽  
Alessandro Calamai ◽  
Massimo Furi ◽  
Maria Patrizia Pera
Keyword(s):  
Hilbert Spaces ◽  
Eigenvalue Problems ◽  
Eigenvalues And Eigenvectors ◽  
Index Zero ◽  
Global Continuation ◽  
Banach Manifolds ◽  
Unperturbed Problem ◽  
Fredholm Maps

We study the persistence of eigenvalues and eigenvectors of perturbed eigenvalue problems in Hilbert spaces. We assume that the unperturbed problem has a nontrivial kernel of odd dimension and we prove a Rabinowitz-type global continuation result. The approach is topological, based on a notion of degree for oriented Fredholm maps of index zero between real differentiable Banach manifolds.

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