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 ( T ) = 8 δ = where d 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 ( 8 ) T = TE ( 8 ) , o ( T ) = 8 δ = where d is an arbitrary spectral set of T and where ...
Page 933
... spectra . The spectral sets of von Neumann . If T is a bounded linear operator in a Hilbert space , then von Neumann [ 3 ] defines a closed set S of the complex sphere to be a spectral set of T if f ( T ) exists and \ f ( T ) | ≤1 ...
... spectra . The spectral sets of von Neumann . If T is a bounded linear operator in a Hilbert space , then von Neumann [ 3 ] defines a closed set S of the complex sphere to be a spectral set of T if f ( T ) exists and \ f ( T ) | ≤1 ...
Page 993
... spectral set consisting of the single point m then , for some complex number a , q ( x ) = x [ x , m ] for almost all x in R. PROOF . In view of Lemma 11 ( d ) it suffices to prove the theorem in the case m = 0. In this case the ...
... spectral set consisting of the single point m then , for some complex number a , q ( x ) = x [ x , m ] for almost all x in R. PROOF . In view of Lemma 11 ( d ) it suffices to prove the theorem in the case m = 0. In this case 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 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 PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero