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 ...
... 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 ...
Page 907
... non - negative real axis respectively . PROOF . If N is a bounded normal operator then , by Corollary IX.3.15 , NN ... 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 IX.3.15 , NN ... 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 ...
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BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero