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 L2 ( 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 L2 ( R ) , which fact was ...
Page 1160
... Plancherel's theorem { Tff Є X } is dense in L2 ( R ) and hence { tƒ · tg | f , g = K } is dense in L1 ( R ) . Now let f , g be in and vanish outside of a compact set C with C = -C . Then if F Plancherel's theorem = TfTg we have by 3.17 ...
... Plancherel's theorem { Tff Є X } is dense in L2 ( R ) and hence { tƒ · tg | f , g = K } is dense in L1 ( R ) . Now let f , g be in and vanish outside of a compact set C with C = -C . Then if F Plancherel's theorem = TfTg we have by 3.17 ...
Page 1666
... Plancherel's theorem . Next suppose that k≥0 . Suppose that || F ( x ) < ∞o . Then , by what has been proved and by Lemma 40 ( ii ) , F is in H ( C ) and | F | ( x ) ≤ A || F || ( x ) , where A is a finite constant depending only on ...
... Plancherel's theorem . Next suppose that k≥0 . Suppose that || F ( x ) < ∞o . Then , by what has been proved and by Lemma 40 ( ii ) , F is in H ( C ) and | F | ( x ) ≤ A || F || ( x ) , where A is a finite constant depending only on ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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 subspace Suppose T₁ T₂ theory To(t topology transform unique unitary vanishes vector zero