Linear Operators: Spectral theory |
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Page 1155
... theorem enables us to refer to the Haar measure on the product group R × R rather than a Haar measure . The Haar ... Plancherel's theorem asserts that the set of charac- ters forms a complete orthonormal set in L¿ ( R ) , which fact was ...
... theorem enables us to refer to the Haar measure on the product group R × R rather than a Haar measure . The Haar ... Plancherel's theorem asserts that the set of charac- ters forms a complete orthonormal set in L¿ ( R ) , which fact was ...
Page 1159
... Plancherel's theorem we have √k [ x , m ] tf ( m ) τf ( m ) μ ( dm ) = [ μ¤ † ,, ( m ) τf ( m ) μ ( dm ) = √ R ... Theorem 3.16 that h vanishes almost everywhere on R and hence h│2 = 0 , which contradicts the Plancherel theorem . To ...
... Plancherel's theorem we have √k [ x , m ] tf ( m ) τf ( m ) μ ( dm ) = [ μ¤ † ,, ( m ) τf ( m ) μ ( dm ) = √ R ... Theorem 3.16 that h vanishes almost everywhere on R and hence h│2 = 0 , which contradicts the Plancherel theorem . To ...
Page 1160
... Plancherel's theorem { Tff ex } is dense in L ( R ) and hence { rf • 7g | f , g € K } is dense in L1 ( ( R ) . Now let f , g be in K and vanish outside of a compact TfTg we have by 3.17 and = -C . Then if F set C with C Plancherel's ...
... Plancherel's theorem { Tff ex } is dense in L ( R ) and hence { rf • 7g | f , g € K } is dense in L1 ( ( R ) . Now let f , g be in K and vanish outside of a compact TfTg we have by 3.17 and = -C . Then if F set C with C Plancherel's ...
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