Linear Operators: Spectral operators |
From inside the book
Results 1-3 of 91
Page 1934
... seen that x * x ( ) = 0 for all έ and all x * x * . Hence , by Corollary II.3.14 , x ( § ) = 0 and thus x = ( ¿ I — T ) x ( § ) = 0 . Q.E.D. → 4 THEOREM . Let T be a bounded spectral operator with resolution of the identity E , and let ...
... seen that x * x ( ) = 0 for all έ and all x * x * . Hence , by Corollary II.3.14 , x ( § ) = 0 and thus x = ( ¿ I — T ) x ( § ) = 0 . Q.E.D. → 4 THEOREM . Let T be a bounded spectral operator with resolution of the identity E , and let ...
Page 2163
... seen from Corollary XV.3.7 that F ( $ ) also commutes with the projec- tions in the range of E , that is , ( iii ) F ( ¿ ) E ( 0 ) = E ( 0 ) F ( § ) , for every Borel set σ . If ξερ ( Τ ) , = { σ1 , ... , σn } , π ' = { 01 , ... , o'n ...
... seen from Corollary XV.3.7 that F ( $ ) also commutes with the projec- tions in the range of E , that is , ( iii ) F ( ¿ ) E ( 0 ) = E ( 0 ) F ( § ) , for every Borel set σ . If ξερ ( Τ ) , = { σ1 , ... , σn } , π ' = { 01 , ... , o'n ...
Page 2479
... seen to be compact by use of Corollary VI.5.5 , and the operator ( iI - H ) -1VE ( e ) = ( ¿ I — H ) ̄1V2QE ( e ) = ( iI — H2 ) ̄1V2 ( E ( e ) | H1 ) Q is seen to belong to the trace class by Lemma XI.9.9 ( d ) . Finally , we have V2 ...
... seen to be compact by use of Corollary VI.5.5 , and the operator ( iI - H ) -1VE ( e ) = ( ¿ I — H ) ̄1V2QE ( e ) = ( iI — H2 ) ̄1V2 ( E ( e ) | H1 ) Q is seen to belong to the trace class by Lemma XI.9.9 ( d ) . Finally , we have V2 ...
Contents
SPECTRAL OPERATORS | 1924 |
14 | 1983 |
Sufficient Conditions | 2134 |
Copyright | |
20 other sections not shown
Other editions - View all
Common terms and phrases
A₁ adjoint operator algebra of projections Amer analytic arbitrary asymptotic B₁ Banach space Boolean algebra Borel set boundary conditions bounded Borel function bounded linear operator bounded operator commuting compact complete Boolean algebra complex numbers complex plane continuous functions converges Corollary countably additive Definition denote differential operator Doklady Akad domain E₁ eigenvalues elements equation exists finite number follows from Lemma follows from Theorem follows immediately formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis identity inequality inverse L₁ Lebesgue Math multiplicity Nauk SSSR norm operators in Hilbert perturbation PROOF properties prove quasi-nilpotent resolution Russian satisfies scalar type operator scalar type spectral Section sequence shows spectral measure spectral operator spectral theory spectrum subset Suppose trace class type spectral operator unbounded uniformly bounded vector zero