Linear Operators: Spectral theory |
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Page 870
... let Q , be the closed disc { a } in the complex plane and let Q = Prex Q z be the Cartesian product of all such discs . Since | x ( M ) | ≤ x it is seen that x ( M ) is in Q , xex , and thus each M in M determines a point q in Q with q ...
... let Q , be the closed disc { a } in the complex plane and let Q = Prex Q z be the Cartesian product of all such discs . Since | x ( M ) | ≤ x it is seen that x ( M ) is in Q , xex , and thus each M in M determines a point q in Q with q ...
Page 1406
... q ( t ) be any real formally self adjoint second order formal differential operator defined on the interval ( -∞ , ∞ ) . Let p ( t ) > 0 for t e I , and let q ( t ) be bounded below . Then τ has no boundary values either at + ∞ or ...
... q ( t ) be any real formally self adjoint second order formal differential operator defined on the interval ( -∞ , ∞ ) . Let p ( t ) > 0 for t e I , and let q ( t ) be bounded below . Then τ has no boundary values either at + ∞ or ...
Page 1678
... ( q ) + āF ( ŷ ) . PROOF . Let & be a second function in Co ( I ) such that y ( x ) = 1 for x in a neighborhood of K1 ... ( q ) + āF ( ŷ ) by Definition 51 and the first paragraph of the present proof . Q.E.D. The following lemma will be ...
... ( q ) + āF ( ŷ ) . PROOF . Let & be a second function in Co ( I ) such that y ( x ) = 1 for x in a neighborhood of K1 ... ( q ) + āF ( ŷ ) by Definition 51 and the first paragraph of the present proof . Q.E.D. The following lemma will be ...
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