Linear Operators: Spectral theory |
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Page 931
... Hilbert space can be extended ( in some sense ) to an operator B in a Hilbert space K , containing H , in such a way that B has properties not possessed by A. One possible type of extension is the following : if t is a Hilbert space ...
... Hilbert space can be extended ( in some sense ) to an operator B in a Hilbert space K , containing H , in such a way that B has properties not possessed by A. One possible type of extension is the following : if t is a Hilbert space ...
Page 932
... Hilbert space § satis- fying F ( $ ) = 0 and F ( S ) = I. Then there exists a Hilbert space 25 and a self adjoint projection valued additive ( resp . weakly countable additive ) function E on Σ to B ( A ) such that if P is the self ...
... Hilbert space § satis- fying F ( $ ) = 0 and F ( S ) = I. Then there exists a Hilbert space 25 and a self adjoint projection valued additive ( resp . weakly countable additive ) function E on Σ to B ( A ) such that if P is the self ...
Page 1180
... Hilbert space . Therefore , Corollary 23 generalizes , with hardly any change in its proof , to the space of functions ƒ with values in any space L , ( § ) , § denoting an arbitrary Hilbert space . Next , it may be noted that Lemma 24 ...
... Hilbert space . Therefore , Corollary 23 generalizes , with hardly any change in its proof , to the space of functions ƒ with values in any space L , ( § ) , § denoting an arbitrary Hilbert space . Next , it may be noted that Lemma 24 ...
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