Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 907
... non - negative then , by Corollary 2.7 , ( Ax , x ) = So ( A ) λ ( E ( dλ ) x , x ) , which shows that the operator A is positive . Conversely , if an open interval d of negative numbers inter- sects the spectrum then , by Lemma 3.3 ( i ) ...
... non - negative then , by Corollary 2.7 , ( Ax , x ) = So ( A ) λ ( E ( dλ ) x , x ) , which shows that the operator A is positive . Conversely , if an open interval d of negative numbers inter- sects the spectrum then , by Lemma 3.3 ( i ) ...
Page 939
... non - negative regular countably additive set function v on Σ with v ( G ) = 1 and v ( Es ) = v ( E ) for E in Σ and s in G. Let v1 be any such function and let μ be any non - negative regular measure on Σ with μ1 ( G ) = 1 and μ1 ( 8E ) ...
... non - negative regular countably additive set function v on Σ with v ( G ) = 1 and v ( Es ) = v ( E ) for E in Σ and s in G. Let v1 be any such function and let μ be any non - negative regular measure on Σ with μ1 ( G ) = 1 and μ1 ( 8E ) ...
Page 1254
... non- negative symmetric operator in . By Theorem 5.2 it has a non- negative self adjoint extension S1 . If E ( - ) denotes the resolution of the identity of S1 , it follows from the proof of Theorem 1 that m1 = √ ° o ̧t " μ ( dt ) , n ...
... non- negative symmetric operator in . By Theorem 5.2 it has a non- negative self adjoint extension S1 . If E ( - ) denotes the resolution of the identity of S1 , it follows from the proof of Theorem 1 that m1 = √ ° o ̧t " μ ( dt ) , n ...
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