Linear Operators: Spectral operators |
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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 ...
... 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 ...
Page 1363
... formula 1 E ( ( 1 , 2 ) ) = lim lim Σπί 21 + 8 + 0-3 0-8 [ R ( λ — iɛ ; T ) —R ( λ + iɛ ; T ) ] ƒdλ . The problem we face is that of passing from this latter formula in- volving the resolvent to a formula involving the individual terms ...
... formula 1 E ( ( 1 , 2 ) ) = lim lim Σπί 21 + 8 + 0-3 0-8 [ R ( λ — iɛ ; T ) —R ( λ + iɛ ; T ) ] ƒdλ . The problem we face is that of passing from this latter formula in- volving the resolvent to a formula involving the individual terms ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 Haar measure 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 operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology unique unitary vanishes vector zero