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 f on R such that f1 < 1 and ( τf ) ( m ) 0. Let a = | ( Tf ) ( m 。) | so that 0 < x < 1 and let U be a ...
... neighborhood of h By IV.8.19 the integrable continuous functions on R are dense in L1 ( R ) so there is a continuous function f on R such that f1 < 1 and ( τf ) ( m ) 0. Let a = | ( Tf ) ( m 。) | so that 0 < x < 1 and let U be a ...
Page 1678
... neighborhood of K1 . Then yq - ŵp vanishes in a neigh- borhood of K C ( F ) , and vanishes in a neighborhood of C ( F ) -K since o vanishes in the complement of K. Hence yp - p vanishes in a neighborhood of C ( F ) , so that F ( pp ) F ...
... neighborhood of K1 . Then yq - ŵp vanishes in a neigh- borhood of K C ( F ) , and vanishes in a neighborhood of C ( F ) -K since o vanishes in the complement of K. Hence yp - p vanishes in a neighborhood of C ( F ) , so that F ( pp ) F ...
Page 1734
... neighborhood of q chosen so small that BU , CE , and so that there exists a mapping of U1 onto the unit spherical neighborhood V of the origin such that ( i ) is one - to - one , is infinitely often differentiable , and q - 1 is ...
... neighborhood of q chosen so small that BU , CE , and so that there exists a mapping of U1 onto the unit spherical neighborhood V of the origin such that ( i ) is one - to - one , is infinitely often differentiable , and q - 1 is ...
Contents
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