Linear Operators, Part 2 |
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Page 891
... set function E. In the present chapter we shall only integrate bounded functions f and so the following discussion of the integral will be restricted to that case . Let be a field of subsets of a set S and let E be a function which maps ...
... set function E. In the present chapter we shall only integrate bounded functions f and so the following discussion of the integral will be restricted to that case . Let be a field of subsets of a set S and let E be a function which maps ...
Page 894
... set functions whose values on a set oЄ _g ( 1 ) E ( dt ) . respectively . The integral Ss f ( s ) E ( ds are E ( od ) , 8 ) is itself a bounded additive set function for 8 in 2 , and thus the integral of a function g in B ( S , E ) with ...
... set functions whose values on a set oЄ _g ( 1 ) E ( dt ) . respectively . The integral Ss f ( s ) E ( ds are E ( od ) , 8 ) is itself a bounded additive set function for 8 in 2 , and thus the integral of a function g in B ( S , E ) with ...
Page 1899
... functions , defi- nition , IV.2.22 ( 242 ) set function . ( See Continuous set function and Set function ) space of , additional properties , IV.15 ( 378 ) definition , IV.2.22 ( 242 ) remarks concerning , ( 392 ) study of , IV.12.3 ...
... functions , defi- nition , IV.2.22 ( 242 ) set function . ( See Continuous set function and Set function ) space of , additional properties , IV.15 ( 378 ) definition , IV.2.22 ( 242 ) remarks concerning , ( 392 ) study of , IV.12.3 ...
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 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 real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero