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 g ( ) into the vector - valued function whose nth component has the Fourier transform h ( ) defined by ( 61 ) hn ( § ) = gn ( § ) , 2 ′′ < | § | < 2 ′′ +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 ) hn ( § ) = gn ( § ) , 2 ′′ < | § | < 2 ′′ +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 . 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 . Then if x is in D ( T ) , we have | ( T + iI ) x | 2 = ( Tx , Tx ) i ( x , Tx ) ...
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