Linear Operators: Spectral theory |
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Page 892
... simple functions converges in B ( S , E ) to then the sequence { fs fn ( s ) E ( ds ) } of integrals converges , and its limit depends only upon f and not upon the particular sequence { f } used to approximate f . Thus we may define the ...
... simple functions converges in B ( S , E ) to then the sequence { fs fn ( s ) E ( ds ) } of integrals converges , and its limit depends only upon f and not upon the particular sequence { f } used to approximate f . Thus we may define the ...
Page 1218
... functions , a well known theorem of Lusin . 17 LEMMA . Let u be a finite positive regular measure on the Borel sets ... simple function has the desired property . Since μ ( R ) < ∞ every μ - measurable function f is the limit in u - measure ...
... functions , a well known theorem of Lusin . 17 LEMMA . Let u be a finite positive regular measure on the Borel sets ... simple function has the desired property . Since μ ( R ) < ∞ every μ - measurable function f is the limit in u - measure ...
Page 1905
... functions , as an oper- ator in L , ( R ) , XI.3.3 ( 954 ) definition , VIII.1.23 ( 633 ) inequalities concerning ... simple functions in L ,, 1 ≤ p < ∞ , III.3.8 ( 125 ) nowhere dense set , I.6.11 ( 21 ) Density of the natural ...
... functions , as an oper- ator in L , ( R ) , XI.3.3 ( 954 ) definition , VIII.1.23 ( 633 ) inequalities concerning ... simple functions in L ,, 1 ≤ p < ∞ , III.3.8 ( 125 ) nowhere dense set , I.6.11 ( 21 ) Density of the natural ...
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