Stability indices for constrained self-adjoint operators
نویسندگان
چکیده
منابع مشابه
Stability indices for constrained self-adjoint operators
A wide class of problems in the study of the spectral and orbital stability of dispersive waves in Hamiltonian systems can be reduced to understanding the so-called “energy spectrum,” that is the spectrum of the second variation of the Hamiltonian evaluated at the wave shape, which is constrained to act on a closed subspace of the underlying Hilbert space. We present a substantially simplified ...
متن کاملSpectral Theory for Compact Self-Adjoint Operators
This agrees with the definition of the spectrum in the matrix case, where the resolvent set comprises all complex numbers that are not eigenvalues. In terms of its spectrum, we will see that a compact operator behaves like a matrix, in the sense that its spectrum is the union of all of its eigenvalues and 0. We begin with the eigenspaces of a compact operator. We start with two lemmas that we w...
متن کاملSpectral Theorem for Self-adjoint Linear Operators
Let V be a finite-dimensional vector space, either real or complex, and equipped with an inner product 〈· , ·〉. Let A : V → V be a linear operator. Recall that the adjoint of A is the linear operator A : V → V characterized by 〈Av, w〉 = 〈v, Aw〉 ∀v, w ∈ V (0.1) A is called self-adjoint (or Hermitian) when A = A. Spectral Theorem. If A is self-adjoint then there is an orthonormal basis (o.n.b.) o...
متن کاملNon-self-adjoint Differential Operators
We describe methods which have been used to analyze the spectrum of non-self-adjoint differential operators, emphasizing the differences from the self-adjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely related to a high degree of instability of the eigenvalues under small perturbations of the opera...
متن کاملAdjoints and Self-Adjoint Operators
Let V and W be real or complex finite dimensional vector spaces with inner products 〈·, ·〉V and 〈·, ·〉W , respectively. Let L : V → W be linear. If there is a transformation L∗ : W → V for which 〈Lv,w〉W = 〈v, Lw〉V (1) holds for every pair of vectors v ∈ V and w in W , then L∗ is said to be the adjoint of L. Some of the properties of L∗ are listed below. Proposition 1.1. Let L : V →W be linear. ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 2012
ISSN: 0002-9939,1088-6826
DOI: 10.1090/s0002-9939-2011-10943-2