Linear Operators: Spectral theory |
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Page 889
... spectrum of an operator is always closed ( IX.1.5 ) , every set in the domain of a spectral measure satisfying ( iii ) is necessarily an open and closed subset of o ( T ) and thus a spectral set . However , in order to reduce the study ...
... spectrum of an operator is always closed ( IX.1.5 ) , every set in the domain of a spectral measure satisfying ( iii ) is necessarily an open and closed subset of o ( T ) and thus a spectral set . However , in order to reduce the study ...
Page 1187
... operator ( λI − T ) −1 . The spectrum o ( T ) of T is the complement of the resolvent set p ( T ) . The point spectrum o , ( T ) , the continuous spectrum σ ( T ) , and the residual spectrum o , ( T ) are defined just as they were in ...
... operator ( λI − T ) −1 . The spectrum o ( T ) of T is the complement of the resolvent set p ( T ) . The point spectrum o , ( T ) , the continuous spectrum σ ( T ) , and the residual spectrum o , ( T ) are defined just as they were in ...
Page 1538
... spectral theory of ordinary differential operators . The present section is ... Operator , I ; ( G ) The Sturm - Liouville Operator , II ; ( H ) The Sturm ... operator on the interval 1538 XIII . ORDINARY DIFFERENTIAL OPERATORS XIII.9.A1.
... spectral theory of ordinary differential operators . The present section is ... Operator , I ; ( G ) The Sturm - Liouville Operator , II ; ( H ) The Sturm ... operator on the interval 1538 XIII . ORDINARY DIFFERENTIAL OPERATORS XIII.9.A1.
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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 PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T unique unitary vanishes vector zero