Linear Operators, Part 2 |
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Page 1040
... analytic even at 2λm . It will now be shown that y1⁄2 ( λ ) = AN E ( Am ; T ) * R ( A ; T ) * y vanishes which will prove that y ( 2 ) is analytic at all the points λ = m , so that y ( 2 ) can only fail to be analytic at the point 2 0 ...
... analytic even at 2λm . It will now be shown that y1⁄2 ( λ ) = AN E ( Am ; T ) * R ( A ; T ) * y vanishes which will prove that y ( 2 ) is analytic at all the points λ = m , so that y ( 2 ) can only fail to be analytic at the point 2 0 ...
Page 1102
... analytic function , it follows that det ( I + zT ) is analytic if -zo ( 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 -zo ( 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 ...
Page 1379
... analytic for j > k , it follows from Theorem 18 that p ,, ( e ) = 0 for j > k and each Borel set e with compact ... analytic functions defined on a neighborhood of λ . PROOF . It is clear from Theorem 18 that if 0 ; can be extended to be ...
... analytic for j > k , it follows from Theorem 18 that p ,, ( e ) = 0 for j > k and each Borel set e with compact ... analytic functions defined on a neighborhood of λ . PROOF . It is clear from Theorem 18 that if 0 ; can be extended to be ...
Contents
BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 coefficients compact operator complex numbers constant 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 prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology unique unitary vanishes vector zero