Linear Operators: Spectral theory |
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Page 949
... seen ( Theorem 2 ) that AP is isometric and isomorphic with C ( S ) , where S is a compact Abelian group , and also ( Lemma 3 ) that the con- tinuous characters of S are of the form ei . By Theorem 1.6 , the set of continuous characters ...
... seen ( Theorem 2 ) that AP is isometric and isomorphic with C ( S ) , where S is a compact Abelian group , and also ( Lemma 3 ) that the con- tinuous characters of S are of the form ei . By Theorem 1.6 , the set of continuous characters ...
Page 1037
... seen that the function ( T ) is analytic for λ 0 and vanishes only for λ in σ ( T ) . It remains to show that if 2 0 , then ( T ) is continuous in T relative to the Hilbert - Schmidt norm in HS . To do this let { T } be a sequence in HS ...
... seen that the function ( T ) is analytic for λ 0 and vanishes only for λ in σ ( T ) . It remains to show that if 2 0 , then ( T ) is continuous in T relative to the Hilbert - Schmidt norm in HS . To do this let { T } be a sequence in HS ...
Page 1154
... seen from Corollary III.11.6 , is a consequence of the assertion that ( ii ) 2 ( 2 ) ( AB ) = cλ ( A ) λ ( B ) , Α , Β Ε Σ . Thus we shall endeavor to establish ( ii ) . For every E in Σ ( 2 ) let μ ( E ) = 2 ( 2 ) ( hE ) where h is the ...
... seen from Corollary III.11.6 , is a consequence of the assertion that ( ii ) 2 ( 2 ) ( AB ) = cλ ( A ) λ ( B ) , Α , Β Ε Σ . Thus we shall endeavor to establish ( ii ) . For every E in Σ ( 2 ) let μ ( E ) = 2 ( 2 ) ( hE ) where h is the ...
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