Linear Operators: Spectral operators |
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Page 888
... spectral measure E defined on the family of spectral sets of T. This spectral measure is also related ( VII.3.20 ) to T by the equations ( iii ) E ( S ) TTE ( 8 ) , σ ( Ts ) = 8 8 = = where & is an arbitrary spectral set of T and where ...
... spectral measure E defined on the family of spectral sets of T. This spectral measure is also related ( VII.3.20 ) to T by the equations ( iii ) E ( S ) TTE ( 8 ) , σ ( Ts ) = 8 8 = = where & is an arbitrary spectral set of T and where ...
Page 889
... spectral measure satisfying ( iii ) is necessarily an open ... spectral set . However , in order to reduce the study of T to its study on invariant subspaces in which it has a smaller spectrum it is quite sufficient to find a spectral measure ...
... spectral measure satisfying ( iii ) is necessarily an open ... spectral set . However , in order to reduce the study of T to its study on invariant subspaces in which it has a smaller spectrum it is quite sufficient to find a spectral measure ...
Page 897
... spectral measure and , in particular , all of the projections E ( 8 ) commute . It follows then from ( iii ) that the projections E ( 8 ) also commute with T ( f ) and this completes the proof of the theorem . Q.E.D. 3 COROLLARY . The ...
... spectral measure and , in particular , all of the projections E ( 8 ) commute . It follows then from ( iii ) that the projections E ( 8 ) also commute with T ( f ) and this completes the proof of the theorem . Q.E.D. 3 COROLLARY . The ...
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BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 Haar measure 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 operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology unique unitary vanishes vector zero