Algebraic Theories in Functional Analysis.
Spectral theorem
An Elementary Treatment of Hilbert Spaces. When are two Banach spaces isomorphic? The spectral theorems form a cornerstone of functional analysis. They are a vast generalization to infinite-dimensional Hilbert spaces of a basic result in linear algebra: There is a caveat, though: The spectral theorem does not say that for every selfadjoint A A there is a basis so that A A has a diagonal matrix with respect to it.
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It should be noted of the works of J. Marcus [3] with the proviso that operators are bounded if and operators are completely continuous if. Through are designated the sequence of the characteristic values of the operator in order of increasing various modules taking into account their multiplicities. In the work [6] it is proved the summation of the series on eigen and associated vectors of the operator pencil by the method of Abel.
The work [5] is devoted also to the questions of multiple summation of series on eigen and associated vectors of operator pencil under the conditions that the resolvent of operator pencil on closed expanding indefinitely contours uniformly bounded. Below it is presented a theorem asserting on multiple summation over the root subspaces of the operator pencil , in other words, the possibility of multiple expansions with brackets on eigen and associated vectors of the operator.
Let , in 1 be completely continuous linear operators, acting in a separable Hilbert space.
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The operator pencil 1 is known in the spectral theory of operators as a pencil of Keldysh. There is an associated vector to the eigenvector if the following series of equations. Linearly-independent elements 4 form a chain of eigen and associated vectors of 1. Under -multiple completeness of eigen and associated vectors of the operator pencil in space is understood the possibility of approaching any element of space by linear combinations of elements , respectively, with the same coefficients independent of the indices of the elements.
If at least for one point of the operator invertible, then the set of eigen values of the pencil consists of isolated points of finite algebraic multiplicity [1]. The study of the spectral properties of the equation in a Hilbert space is reduced to the study of the spectral properties of the equation in the direct sum of copies of the space.
Operator is a normal completely continuous , characteristic values of lie on rays emanated from origin, norms of characteristic values of operators and coincide. Suppose that the following conditions are satisfied: Then we have the multiple basis of eigen and associated vectors with brackets in the range of the operator.
If then the multiple completeness of the system of eigen and associated vectors of the operator in the space takes place. We evaluate the resolvent - operator for the values. Suppose the estimate holds for all. Finite induction then finishes the proof. The spectral theorem holds also for symmetric maps on finite-dimensional real inner product spaces, but the existence of an eigenvector does not follow immediately from the fundamental theorem of algebra.
To prove this, consider A as a Hermitian matrix and use the fact that all eigenvalues of a Hermitian matrix are real. If one chooses the eigenvectors of A as an orthonormal basis, the matrix representation of A in this basis is diagonal. Equivalently, A can be written as a linear combination of pairwise orthogonal projections, called its spectral decomposition.
Note that the definition does not depend on any choice of specific eigenvectors. The spectral decomposition is a special case of both the Schur decomposition and the singular value decomposition. The spectral theorem extends to a more general class of matrices. Let A be an operator on a finite-dimensional inner product space.
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One can show that A is normal if and only if it is unitarily diagonalizable. Therefore, T must be diagonal since a normal upper triangular matrix is diagonal see normal matrix.
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The converse is obvious. In other words, A is normal if and only if there exists a unitary matrix U such that. Then, the entries of the diagonal of D are the eigenvalues of A. The column vectors of U are the eigenvectors of A and they are orthonormal.
Spectral theorem - Wikipedia
Unlike the Hermitian case, the entries of D need not be real. In the more general setting of Hilbert spaces, which may have an infinite dimension, the statement of the spectral theorem for compact self-adjoint operators is virtually the same as in the finite-dimensional case. Suppose A is a compact self-adjoint operator on a real or complex Hilbert space V. Then there is an orthonormal basis of V consisting of eigenvectors of A.
As for Hermitian matrices, the key point is to prove the existence of at least one nonzero eigenvector. One cannot rely on determinants to show existence of eigenvalues, but one can use a maximization argument analogous to the variational characterization of eigenvalues. If the compactness assumption is removed, it is not true that every self-adjoint operator has eigenvectors.
The next generalization we consider is that of bounded self-adjoint operators on a Hilbert space. Such operators may have no eigenvalues: A delta-function, however, is not a normalizable function; that is, it is not actually in the Hilbert space L 2 [0, 1].