Linear Operators: Spectral theory |
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Page 995
... containing the remainder of o ( f * q ) . It follows from Lemma 12 that the set ( h * f * q ) contains at most the single point m 。 and hence , from Theorem 16 and Lemma 3.1 ( d ) , that there is a number a with ( h * f * ) ( x ) ax ...
... containing the remainder of o ( f * q ) . It follows from Lemma 12 that the set ( h * f * q ) contains at most the single point m 。 and hence , from Theorem 16 and Lemma 3.1 ( d ) , that there is a number a with ( h * f * ) ( x ) ax ...
Page 996
... contains no interior point of o ( p ) . Hence o ( ƒ * q ) is a closed subset of the boundary of o ( q ) . Since ƒ * q = 0 it follows from Lemma 11 ( a ) that o ( ƒ * q ) is not void . Thus , by hypothesis , o ( f ) contains an isolated ...
... contains no interior point of o ( p ) . Hence o ( ƒ * q ) is a closed subset of the boundary of o ( q ) . Since ƒ * q = 0 it follows from Lemma 11 ( a ) that o ( ƒ * q ) is not void . Thus , by hypothesis , o ( f ) contains an isolated ...
Page 1700
... contains Σ , then the conditions 0 ( a , ( ) ) ' f ( x ) = 0 , ΤΕΣ . 0 ≤ j ≤k - 1 , and ( a , ( E ) ) ' ( f \ L ) ( x ) = 0 , = ΜΕΣ , 0 ≤ j ≤k - 1 , are equivalent ; ( ii ) if I is a subdomain of I whose boundary contains a smooth ...
... contains Σ , then the conditions 0 ( a , ( ) ) ' f ( x ) = 0 , ΤΕΣ . 0 ≤ j ≤k - 1 , and ( a , ( E ) ) ' ( f \ L ) ( x ) = 0 , = ΜΕΣ , 0 ≤ j ≤k - 1 , are equivalent ; ( ii ) if I is a subdomain of I whose boundary contains a smooth ...
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BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero