Linear Operators: Spectral theory |
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Page 893
... bounded E - measurable functions on S into the B * -algebra of bounded operators on Hilbert space . Returning now to the general integral ff ( s ) E ( ds ) where E is merely a bounded additive operator valued set function , we observe ...
... bounded E - measurable functions on S into the B * -algebra of bounded operators on Hilbert space . Returning now to the general integral ff ( s ) E ( ds ) where E is merely a bounded additive operator valued set function , we observe ...
Page 900
... bounded 2 - measurable scalar functions on S in such a way that each equivalence class consists of all E - measurable functions which differ from some bounded E - measurable function only on a set of E measure zero . That is , EB ( S ...
... bounded 2 - measurable scalar functions on S in such a way that each equivalence class consists of all E - measurable functions which differ from some bounded E - measurable function only on a set of E measure zero . That is , EB ( S ...
Page 1452
... bounded , ( Tx , x ) is bounded below . Conversely , suppose that for each n , en = ( -∞ , —n ) non - void . By Theorem XII.2.9 ( b ) , E ( en ) this fact , choose an a , such that E ( e ) x , Then , by Theorem XII.2.6 , -n = x = D ( T ) ...
... bounded , ( Tx , x ) is bounded below . Conversely , suppose that for each n , en = ( -∞ , —n ) non - void . By Theorem XII.2.9 ( b ) , E ( en ) this fact , choose an a , such that E ( e ) x , Then , by Theorem XII.2.6 , -n = x = D ( T ) ...
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 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 open set 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 tr(T unique unitary vanishes vector zero