Linear Operators: Spectral theory |
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Page 890
... formula ( vi ) , but in this situation it is necessary to define the integral appearing in ( vi ) and to define the algebra of scalar functions f to which the formula may be applied . One class of scalar functions f , other than ...
... formula ( vi ) , but in this situation it is necessary to define the integral appearing in ( vi ) and to define the algebra of scalar functions f to which the formula may be applied . One class of scalar functions f , other than ...
Page 1089
... formula " for the eigenvalues of a com- pact operator , given as Theorem X.4.3 . Q.E.D. It will be convenient in what follows to adopt the formula of Lemma 2 as a definition of μ , ( T ) in case T is not compact . Note that T = μ1 ( T ) ...
... formula " for the eigenvalues of a com- pact operator , given as Theorem X.4.3 . Q.E.D. It will be convenient in what follows to adopt the formula of Lemma 2 as a definition of μ , ( T ) in case T is not compact . Note that T = μ1 ( T ) ...
Page 1363
... formula = E ( ( 21 , λ2 ) ) f lim lim + 0-3 0-8 1 Σπί 21 + 8 [ R ( 2 - iɛ ; T ) -R ( λ + iɛ ; T ) ] fdλ . The problem we face is that of passing from this latter formula in- volving the resolvent to a formula involving the individual ...
... formula = E ( ( 21 , λ2 ) ) f lim lim + 0-3 0-8 1 Σπί 21 + 8 [ R ( 2 - iɛ ; T ) -R ( λ + iɛ ; T ) ] fdλ . The problem we face is that of passing from this latter formula in- volving the resolvent to a formula involving the individual ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ 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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero