Linear Operators, Part 2 |
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Page 1074
... transform of a function in L1 ( —∞ , is the Fourier transform of a function in L1 ( —∞ , ∞ ) . Show that for 1 p≤2 , λ ( ) F ( ) is the Fourier transform of a function in L „ ( − ∞ , + ∞ ) whenever F is the Fourier transform of a ...
... transform of a function in L1 ( —∞ , is the Fourier transform of a function in L1 ( —∞ , ∞ ) . Show that for 1 p≤2 , λ ( ) F ( ) is the Fourier transform of a function in L „ ( − ∞ , + ∞ ) whenever F is the Fourier transform of a ...
Page 1178
... transform g , ( § ) into the vector - valued function whose nth component has the Fourier transform h ( § ) defined by ( 61 ) h2 ( § ) = g ( § ) , 2 " < | < 2n + 1 , = = 0 , otherwise . By Corollary 24 , M is a bounded linear transformation ...
... transform g , ( § ) into the vector - valued function whose nth component has the Fourier transform h ( § ) defined by ( 61 ) h2 ( § ) = g ( § ) , 2 " < | < 2n + 1 , = = 0 , 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 + iI ) 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 + iI ) x | 2 = ( Tx , Tx ) ‡ i ( x , Tx ) ...
Contents
BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 coefficients compact operator complex numbers constant 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 prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology unique unitary vanishes vector zero