Linear Operators: Spectral theory |
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Page 905
... non - zero vector is orthogonal to every element of B. Hence B is complete and , by Theorem IV.4.13 , B is an orthonormal basis for . The remaining two assertions follow from Definition IV.4.11 and Theorem IV.4.10 . Q.E.D. 5 COROLLARY ...
... non - zero vector is orthogonal to every element of B. Hence B is complete and , by Theorem IV.4.13 , B is an orthonormal basis for . The remaining two assertions follow from Definition IV.4.11 and Theorem IV.4.10 . Q.E.D. 5 COROLLARY ...
Page 907
... non - negative real axis respectively . PROOF . If N is a bounded normal operator then , by Corollary I if and only ... zero ( provided that § is infinite dimensional ) . Moreover , it is seen from Corollary 3.5 that the set of eigenvectors ...
... non - negative real axis respectively . PROOF . If N is a bounded normal operator then , by Corollary I if and only ... zero ( provided that § is infinite dimensional ) . Moreover , it is seen from Corollary 3.5 that the set of eigenvectors ...
Page 1260
... non - zero eigenvalues of T * T are the same as the non - zero eigenvalues of TT * , even as to multiplicity ( the positive square roots of these eigenvalues are sometimes called the characteristic numbers of T ) . 18 Let T be a bounded ...
... non - zero eigenvalues of T * T are the same as the non - zero eigenvalues of TT * , even as to multiplicity ( the positive square roots of these eigenvalues are sometimes called the characteristic numbers of T ) . 18 Let T be a bounded ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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A₁ adjoint extension adjoint operator algebra analytic B-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers continuous function converges Corollary deficiency indices Definition denote dense eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology transform unique unitary vanishes vector zero