Linear Operators: Spectral theory |
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Page 1037
... 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 with || T - T || → 0 . Then if ...
... 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 with || T - T || → 0 . Then if ...
Page 1040
... analytic at all the points λm , so that y ( 2 ) can only fail to be analytic at = the point = 0. To show this , note that ( y2 ( λ ) , x ) = ( ¿ N E ( Ãm ; T ) * R ( Ã ; T ) * y , x ) = 2a ( y , E ( Ãm ; T ) R ( 2 ; T ) x ) . Now it ...
... analytic at all the points λm , so that y ( 2 ) can only fail to be analytic at = the point = 0. To show this , note that ( y2 ( λ ) , x ) = ( ¿ N E ( Ãm ; T ) * R ( Ã ; T ) * y , x ) = 2a ( y , E ( Ãm ; T ) R ( 2 ; T ) x ) . Now it ...
Page 1102
... analytic function , it follows that det ( I + zT ) is analytic if -z - 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 ...
... analytic function , it follows that det ( I + zT ) is analytic if -z - 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 ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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