Linear Operators: Spectral theory |
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Page 873
... set in M of all M , with 2 € 4. To see that Mis dense in M suppose the contrary and let { 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 € 4. To see that Mis dense in M suppose the contrary and let { 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 ) , Ĵ 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 ( f ) ( m ) = ay for every m in Vo , from which it follows that av . = ay ...
... open set V. If f is in L1 ( R ) ~ L2 ( R ) , Ĵ 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 ( f ) ( m ) = ay for every m in Vo , from which it follows that av . = ay ...
Page 1151
... sets in R. We select an open set G1 in R such that F1K1 СG1 , Ğ1 ○ F2 = $ , 1 and then choose an open set H1 such that F2OK1CH1 , H1 ~ ( F1 ~ Ğ1 ) = þ . 1 By induction , choose open sets G and H such that FLOK , CG , F2OK CH n n n 2 n ...
... sets in R. We select an open set G1 in R such that F1K1 СG1 , Ğ1 ○ F2 = $ , 1 and then choose an open set H1 such that F2OK1CH1 , H1 ~ ( F1 ~ Ğ1 ) = þ . 1 By induction , choose open sets G and H such that FLOK , CG , F2OK CH n n n 2 n ...
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