Linear Operators: Spectral theory |
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Page 1033
... analytic and vanishes at = ∞ . It follows immediately that g has the Laurent expansion с g ( 2 ) = = aλ + b + λ + ... in the neighborhood of 2 = ∞ . Consequently , the analytic function g ( λ ) -aλ is analytic for all finite and ...
... analytic and vanishes at = ∞ . It follows immediately that g has the Laurent expansion с g ( 2 ) = = aλ + b + λ + ... in the neighborhood of 2 = ∞ . Consequently , the analytic function g ( λ ) -aλ is analytic for all finite and ...
Page 1040
... analytic at all the points λ λm , so that y ( 2 ) can only fail to be analytic at the point 2 = 0. To show this , note that = ( y2 ( 2 ) , x ) = ( 2a E ( Ã „ ; T ) * R ( Ã ; T ) * y , x ) = 2a ( y , E ( Ã „ ; T ) R ( 2 ; T ) x ) . Now ...
... analytic at all the points λ λm , so that y ( 2 ) can only fail to be analytic at the point 2 = 0. To show this , note that = ( y2 ( 2 ) , x ) = ( 2a E ( Ã „ ; T ) * R ( Ã ; T ) * y , x ) = 2a ( y , E ( Ã „ ; T ) R ( 2 ; T ) x ) . Now ...
Page 1102
... analytic function , it follows that det ( I + zT ) is analytic if --1o ( T ) . Since by ( a ) det ( I + zT ) is bounded , the singularities are removable and ( b ) is proved . Q.E.D. Remark . Since , by the maximum modulus principle , a ...
... analytic function , it follows that det ( I + zT ) is analytic if --1o ( T ) . Since by ( a ) det ( I + zT ) is bounded , the singularities are removable and ( b ) is proved . Q.E.D. Remark . Since , by the maximum modulus principle , a ...
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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