Linear Operators: Spectral theory |
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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 ( 8 ) T = TE ( 8 ) , o ( Ts ) = 8 น = = where 8 is an arbitrary spectral set of T and ...
... 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 ( 8 ) T = TE ( 8 ) , o ( Ts ) = 8 น = = where 8 is an arbitrary spectral set of T and ...
Page 889
... spectral measure which satisfies , instead of ( iii ) , the condition ( iv ) E ( 8 ) T = TE ( 8 ) , ( Ts ) _ Ỗ , where is the closure of 8. As will be seen in the next section , a normal operator T in Hilbert space determines a spectral ...
... spectral measure which satisfies , instead of ( iii ) , the condition ( iv ) E ( 8 ) T = TE ( 8 ) , ( Ts ) _ Ỗ , where is the closure of 8. As will be seen in the next section , a normal operator T in Hilbert space determines a spectral ...
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 ( ƒ ) and this completes the proof of the theorem . Q.E.D. 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 ( ƒ ) and this completes the proof of the theorem . Q.E.D. COROLLARY . The ...
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