Linear Operators: Spectral theory |
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Page 968
... neighborhood of h By IV.8.19 the integrable continuous functions on R are dense in L1 ( R ) so there is a continuous function fon R such that f1 < 1 and ( τf ) ( m ) 0. Let a = | ( tf ) ( mo ) | so that 0 < a < 1 and let U be a neighborhood ...
... neighborhood of h By IV.8.19 the integrable continuous functions on R are dense in L1 ( R ) so there is a continuous function fon R such that f1 < 1 and ( τf ) ( m ) 0. Let a = | ( tf ) ( mo ) | so that 0 < a < 1 and let U be a neighborhood ...
Page 1403
... neighborhood 。 such that W1 ( • , λ ) € L¿ ( a , c ) for μ - almost all 2 € 4 , since A may then be written as a countable union of such neighborhoods A 。. We shall show below that for each e there exists a neighborhood 4 of 2 , an ...
... neighborhood 。 such that W1 ( • , λ ) € L¿ ( a , c ) for μ - almost all 2 € 4 , since A may then be written as a countable union of such neighborhoods A 。. We shall show below that for each e there exists a neighborhood 4 of 2 , an ...
Page 1678
... neighborhood of C ( F ) -K since o vanishes in the complement of K. Hence yo - yo vanishes in a neighborhood of C ( F ) , so that F ( pp ) F ( q ) by Definition 11 . in Co ( I ) with ( x ) = 1 for By Lemma 2.1 , there is a function x in ...
... neighborhood of C ( F ) -K since o vanishes in the complement of K. Hence yo - yo vanishes in a neighborhood of C ( F ) , so that F ( pp ) F ( q ) by Definition 11 . in Co ( I ) with ( x ) = 1 for By Lemma 2.1 , there is a function x in ...
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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