Linear Operators: Spectral theory |
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Page 1174
... valued function and put g ( s ) = h ( s ) x * if s is in e , and g ( s ) = 0 if s is in none of the sets e . Then ( 43 ) = √ g ( s ) f ( s ) ds = [ h ( s ) | f ( s ) \ ds ; hence ( 42 ) follows from the similar well - known equation ...
... valued function and put g ( s ) = h ( s ) x * if s is in e , and g ( s ) = 0 if s is in none of the sets e . Then ( 43 ) = √ g ( s ) f ( s ) ds = [ h ( s ) | f ( s ) \ ds ; hence ( 42 ) follows from the similar well - known equation ...
Page 1178
... valued functions into functions with values in l . It is plain from Plancherel's theorem that is a bounded mapping ... function whose nth component has the Fourier transform ( § ) into the vector - valued function whose nth component has ...
... valued functions into functions with values in l . It is plain from Plancherel's theorem that is a bounded mapping ... function whose nth component has the Fourier transform ( § ) into the vector - valued function whose nth component has ...
Page 1179
... valued function f whose Fourier transform is defined by ( 64 ) f ( E ) = g ( § ) , 2 " < | § | < 2 ′′ +1 . This ... valued function with the Fourier transform f ( § ) into the vector - valued function whose nth component has the Fourier ...
... valued function f whose Fourier transform is defined by ( 64 ) f ( E ) = g ( § ) , 2 " < | § | < 2 ′′ +1 . This ... valued function with the Fourier transform f ( § ) into the vector - valued function whose nth component has the Fourier ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Copyright | |
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Common terms and phrases
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