Linear Operators: Spectral theory |
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Page 1074
... transform of a function in L1 ( —∞ , ∞ ) whenever F ∞ ) . Show that is the Fourier transform of a function in L1 ( —∞ , for 1 ≤ p ≤ 2 , λ ( - ) F ( - ) is the Fourier transform of a function in L ( -∞ , ∞∞ ) whenever F is the ...
... transform of a function in L1 ( —∞ , ∞ ) whenever F ∞ ) . Show that is the Fourier transform of a function in L1 ( —∞ , for 1 ≤ p ≤ 2 , λ ( - ) F ( - ) is the Fourier transform of a function in L ( -∞ , ∞∞ ) whenever F is the ...
Page 1178
... transform ( § ) into the vector - valued function whose nth component has the Fourier transform h ( ) defined by ( 61 ) hn ( 5 ) = gn ( § ) , - 0 , 2 ′′ < | § | < 2 ′′ + 1 , otherwise . By Corollary 24 , M is a bounded linear transformation ...
... transform ( § ) into the vector - valued function whose nth component has the Fourier transform h ( ) defined by ( 61 ) hn ( 5 ) = gn ( § ) , - 0 , 2 ′′ < | § | < 2 ′′ + 1 , otherwise . By Corollary 24 , M is a bounded linear transformation ...
Page 1271
... transform can be used to determine when a symmetric operator has a self adjoint extension . Let T be a sym- metric operator with domain D ( T ) dense in H. Then if x is in D ( T ) , we have | ( T ± il ) x | 2 : * ( Tx , Tx ) i ( x , Tx ) ...
... transform can be used to determine when a symmetric operator has a self adjoint extension . Let T be a sym- metric operator with domain D ( T ) dense in H. Then if x is in D ( T ) , we have | ( T ± il ) x | 2 : * ( Tx , Tx ) i ( x , Tx ) ...
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