On symmetrizing the ultraspherical spectral method for self-adjoint problems
نویسندگان
چکیده
منابع مشابه
Spectral 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 کاملSpectral Theorem for Bounded Self-adjoint Operators
Diagonalization is one of the most important topics one learns in an elementary linear algebra course. Unfortunately, it only works on finite dimensional vector spaces, where linear operators can be represented by finite matrices. Later, one encounters infinite dimensional vector spaces (spaces of sequences, for example), where linear operators can be thought of as ”infinite matrices”. Extendin...
متن کاملSpectral theorem for self-adjoint continuous operators on Hilbert spaces
Let T be a continuous linear map V → V for a (separable) Hilbert space V . Its spectrum σ(T ) is a compact subset of R, so is certainly contained in some finite interval [a, b]. As usual, for a self-adjoint continuous operator S on V , write S ≥ 0 when 〈Sv, v〉 ≥ 0 for all v ∈ V . For self-adjoint S, T , write S ≤ T when T − S ≥ 0. At the outset, with a ≤ −|T |op and b ≥ |T |op, we have, 〈a · v,...
متن کاملThe Spectral Theorem for Self-Adjoint and Unitary Operators
(1.1) (Au, v) = (u, A∗v), u, v ∈ H. We say A is self-adjoint if A = A∗. We say U ∈ L(H) is unitary if U∗ = U−1. More generally, if H is another Hilbert space, we say Φ ∈ L(H,H) is unitary provided Φ is one-to-one and onto, and (Φu, Φv)H = (u, v)H , for all u, v ∈ H. If dim H = n < ∞, each self-adjoint A ∈ L(H) has the property that H has an orthonormal basis of eigenvectors of A. The same holds...
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: Journal of Computational Physics
سال: 2020
ISSN: 0021-9991
DOI: 10.1016/j.jcp.2020.109383