Linear Operators: Spectral theory |
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Page 1037
... analytic for 20 and vanishes only for λ in σ ( T ) . It remains to show that if λ 0 , then îλ ( 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 C is ...
... analytic for 20 and vanishes only for λ in σ ( T ) . It remains to show that if λ 0 , then îλ ( 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 C is ...
Page 1040
... analytic at the point 20. To show this , note that ' m " ( y2 ( λ ) , x ) ( 2N E ( Ãm ; T ) * R ( Ã ; T ) * y , x ) ... analytic everywhere in the plane except at the origin . Suppose that this function is also known to be analytic at the ...
... analytic at the point 20. To show this , note that ' m " ( y2 ( λ ) , x ) ( 2N E ( Ãm ; T ) * R ( Ã ; T ) * y , x ) ... analytic everywhere in the plane except at the origin . Suppose that this function is also known to be analytic at the ...
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
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero