Linear Operators: Spectral theory |
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Page 1174
... valued function and put g ( s ) = h ( s ) x if s is in none of the sets e ;. Then ( 43 ) . . = if s is in e , and g ( s ) = 0 [ 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 none of the sets e ;. Then ( 43 ) . . = if s is in e , and g ( s ) = 0 [ 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 with the Fourier transform f ( § ) into the vector - valued function whose nth component has the Fourier transform fn ( ) defined by ( 65 ) fn ( § ) = f ( § ) , 2 " ≤ § < 2 " +1 , = 0 , otherwise , is a bounded map of L ...
... valued function with the Fourier transform f ( § ) into the vector - valued function whose nth component has the Fourier transform fn ( ) defined by ( 65 ) fn ( § ) = f ( § ) , 2 " ≤ § < 2 " +1 , = 0 , otherwise , is a bounded map of L ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum 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 kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero