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 for ...
... 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 for ...
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 g ( ) 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 g ( ) into the vector - valued function whose nth component has ...
Page 1179
... valued function f whose Fourier transform is defined by ( 64 ) ƒ ( E ) = g ( § ) , 2 " < § < 2n + 1 . This proves ... 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 ) ƒ ( E ) = g ( § ) , 2 " < § < 2n + 1 . This proves ... valued function with the Fourier transform f ( ) into the vector - valued function whose nth component has the Fourier ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ 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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero