Linear Operators: Spectral theory |
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Page 947
... R + U2 ~ R , 1 i where the sum is taken in R and the closure in S. If V , e U1 , then W ( U1 , U2 ) ^ W ( V1 , V2 ) ≥ W ( U1 ^ V1 , U2 ^ V1⁄2 ) and so the family W = { W ( U1 , U2 ) U , e } has the finite intersection property . Since ...
... R + U2 ~ R , 1 i where the sum is taken in R and the closure in S. If V , e U1 , then W ( U1 , U2 ) ^ W ( V1 , V2 ) ≥ W ( U1 ^ V1 , U2 ^ V1⁄2 ) and so the family W = { W ( U1 , U2 ) U , e } has the finite intersection property . Since ...
Page 948
... in R then h ( s ) -s , and so sh ( s ) = 0 for s in a dense set R and thus identically on S. Consequently h ( s ) is the inverse of s , and S is a topological group . Q.E.D. = - 3 LEMMA . The continuous characters of the compact Abelian ...
... in R then h ( s ) -s , and so sh ( s ) = 0 for s in a dense set R and thus identically on S. Consequently h ( s ) is the inverse of s , and S is a topological group . Q.E.D. = - 3 LEMMA . The continuous characters of the compact Abelian ...
Page 1159
... of R into Ề . 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 H = L1 ~ L1⁄2 ( R ) with │H│2 # 0 but such that îH vanishes on × ( R ) . If h ...
... of R into Ề . 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 H = L1 ~ L1⁄2 ( R ) with │H│2 # 0 but such that îH vanishes on × ( R ) . If h ...
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 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 PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero