Linear Operators: Spectral theory |
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Page 873
... set in M of all M , with 2 e 4. To see that MA is dense in M suppose the contrary and let { M || x ̧ ( M ) —x , ( Mo ) ... open . Thus & is continuous . To see that d - 1 is continuous , i.e. , to see that & maps open sets onto open sets ...
... set in M of all M , with 2 e 4. To see that MA is dense in M suppose the contrary and let { M || x ̧ ( M ) —x , ( Mo ) ... open . Thus & is continuous . To see that d - 1 is continuous , i.e. , to see that & maps open sets onto open sets ...
Page 993
... open set V. If ƒ is in L1 ( R ) ~ L2 ( R ) , ƒ 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 = ay . Now let V1 be an ...
... open set V. If ƒ is in L1 ( R ) ~ L2 ( R ) , ƒ 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 = ay . Now let V1 be an ...
Page 1151
... sets in R. We select an open set G1 in R such that F1OK1G1 , Ğ1 ○ F2 = $ , and then choose an open set H1 such that F2OK , CH1 , H1ˆ ( F1 ~ Ğ1 ) = ‡ . By induction , choose open sets G and H , such that n FOK , CG n 2 Ğ ( F2 H1 ...
... sets in R. We select an open set G1 in R such that F1OK1G1 , Ğ1 ○ F2 = $ , and then choose an open set H1 such that F2OK , CH1 , H1ˆ ( F1 ~ Ğ1 ) = ‡ . By induction , choose open sets G and H , such that n FOK , CG n 2 Ğ ( F2 H1 ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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Common terms and phrases
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 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 real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T unique unitary vanishes vector zero