Spectral concentration for self-adjoint operators
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...
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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...
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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...
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The spectral theorem for commuting self-adjoint operators along with the associated functional (or operational) calculus is among the most useful and beautiful results of analysis. It is well known that forming a functional calculus for noncommuting self-adjoint operators is far more problematic. The central result of this paper establishes a rich functional calculus for any finite number of no...
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(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...
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ژورنال
عنوان ژورنال: Pacific Journal of Mathematics
سال: 1967
ISSN: 0030-8730,0030-8730
DOI: 10.2140/pjm.1967.23.377