Linear Operators: Spectral theory |
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Page 873
... set in M of all M , with 2 e 1. To see that Mis dense in M suppose the contrary and let E { M || x¿ ( M ) —x¿ ( Mo ) ... open . Thus & is continuous . To see that 8-1 is continuous , i.e. , to see that & maps open sets onto open sets note ...
... set in M of all M , with 2 e 1. To see that Mis dense in M suppose the contrary and let E { M || x¿ ( M ) —x¿ ( Mo ) ... open . Thus & is continuous . To see that 8-1 is continuous , i.e. , to see that & maps open sets onto open sets note ...
Page 993
... open set V. If f is in L1 ( R ) ~ L2 ( R ) , f vanishes on the complement of V , and f ( m ) 1 for m in an open subset Vo of V , then the above proof shows that ( Pf ) ( m ) = ay for every m in Vo , from which it . follows that = ay ...
... open set V. If f is in L1 ( R ) ~ L2 ( R ) , f vanishes on the complement of V , and f ( m ) 1 for m in an open subset Vo of V , then the above proof shows that ( Pf ) ( m ) = ay for every m in Vo , from which it . follows that = ay ...
Page 1151
... sets in R. We select an open set G1 in R such that FOK1CG , G1 F2 = $ , and then choose an open set H1 such that 1 & , F2 K1 С H1 , н1 ○ ( F1 ~ Ğ1 ) = $ . 2 1 1 , 1 By induction , choose open sets G and H such that F1KCG 1 = F2OK , CH ...
... sets in R. We select an open set G1 in R such that FOK1CG , G1 F2 = $ , and then choose an open set H1 such that 1 & , F2 K1 С H1 , н1 ○ ( F1 ~ Ğ1 ) = $ . 2 1 1 , 1 By induction , choose open sets G and H such that F1KCG 1 = F2OK , CH ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero