Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 873
... set in M of all M , with 2 e A. 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 8-1 is continuous , i.e. , to see that & maps open sets onto open sets ...
... set in M of all M , with 2 e A. 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 8-1 is continuous , i.e. , to see that & maps open sets onto open sets ...
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 ) = xy for every m in Vo , from which it follows that ay 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 ) = xy for every m in Vo , from which it follows that ay ay ...
Page 1151
... sets in R. We select an open set G , in R such that FOKCG , G10 F2 = $ , 1 and then choose an open set H1 such that 1 F2OK1CH1 , H1 ~ ( F1 ~ Ğ1 ) = $ . 2 1 19 1 1 By induction , choose open sets G , and H , such that F1KG , G1 ( F2H ...
... sets in R. We select an open set G , in R such that FOKCG , G10 F2 = $ , 1 and then choose an open set H1 such that 1 F2OK1CH1 , H1 ~ ( F1 ~ Ğ1 ) = $ . 2 1 19 1 1 By induction , choose open sets G , and H , such that F1KG , G1 ( F2H ...
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 Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm 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 transform unique unitary vanishes vector zero