Linear Operators: Spectral theory |
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Page 910
... shows that f g u - almost everywhere . Hence we may define the operator U1 from D1 to L2 ( u ) by placing U1f ( T ) x = f . Clearly U1 is linear and for y g ( T ) x we have , from Corollary 2.8 , = 1 = f ( T ) x , ≈ = ( y , z ) = √or ...
... shows that f g u - almost everywhere . Hence we may define the operator U1 from D1 to L2 ( u ) by placing U1f ( T ) x = f . Clearly U1 is linear and for y g ( T ) x we have , from Corollary 2.8 , = 1 = f ( T ) x , ≈ = ( y , z ) = √or ...
Page 985
... shows that F ( f * g ) = 0 for every g in L ( R ) . It follows from Corollary II.3.13 that f * g is in L for every g in L ( R ) , which shows that 2 is an ideal and thus 2QJ . Conversely let ƒ be in the closed ideal in L1 ( R ) and ...
... shows that F ( f * g ) = 0 for every g in L ( R ) . It follows from Corollary II.3.13 that f * g is in L for every g in L ( R ) , which shows that 2 is an ideal and thus 2QJ . Conversely let ƒ be in the closed ideal in L1 ( R ) and ...
Page 987
... shows that h , is in 2 and proves that L is invariant under translations . The preceding theorem shows that there is a point m 。 in R with h ( m 。) = 0 for every h in 2. Furthermore , since is closed in the L1 ( R ) topology of L ( R ) ...
... shows that h , is in 2 and proves that L is invariant under translations . The preceding theorem shows that there is a point m 。 in R with h ( m 。) = 0 for every h in 2. Furthermore , since is closed in the L1 ( R ) topology of L ( R ) ...
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BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 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 PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero