Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 948
... R is dense in S , and that with on R , we conclude immediately that s 0 = 82 $ 1 and - ✪ $ 1 ( 8283 ) = ( $ 1 → $ 2 ) $ 3 . coincides 8 , 81 82 It remains to be shown that inverse elements under → exist in S. Consider the mapping H ...
... R is dense in S , and that with on R , we conclude immediately that s 0 = 82 $ 1 and - ✪ $ 1 ( 8283 ) = ( $ 1 → $ 2 ) $ 3 . coincides 8 , 81 82 It remains to be shown that inverse elements under → exist in S. Consider the mapping H ...
Page 1042
... R ( λn ; T ) x , z ) N1∞ and hence x - y belongs to ( R ( T ) + ) + 31 = = - -lim ( TR ( 2 ; T ) x , z ) = 0 211∞ R ( T ) . Q.E.D. COROLLARY . Let T be a densely defined unbounded operator in Hilbert space H , with the property that ...
... R ( λn ; T ) x , z ) N1∞ and hence x - y belongs to ( R ( T ) + ) + 31 = = - -lim ( TR ( 2 ; T ) x , z ) = 0 211∞ R ( T ) . Q.E.D. COROLLARY . Let T be a densely defined unbounded operator in Hilbert space H , with the property that ...
Page 1159
... of R into R. R. Next we shall show that ( R ) is dense in the space . If not , then by applying Lemma 4.2 to R , we find that there exists a function HELL ( R ) with H20 but such that tH vanishes on ( R ) . If h = T1H € L ( R ) it ...
... of R into R. R. Next we shall show that ( R ) is dense in the space . If not , then by applying Lemma 4.2 to R , we find that there exists a function HELL ( R ) with H20 but such that tH vanishes on ( R ) . If h = T1H € L ( R ) it ...
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