Linear Operators: Spectral operators |
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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 & be ...
... 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 & be ...
Page 2093
... seen that X ( F ) is a linear manifold in X , but it need not be closed even when F is closed ; an elementary counterexample is given in Colojoară and Foias [ 4 ; p.25 ] . If T is a spectral operator with resolution of the identity E ...
... seen that X ( F ) is a linear manifold in X , but it need not be closed even when F is closed ; an elementary counterexample is given in Colojoară and Foias [ 4 ; p.25 ] . If T is a spectral operator with resolution of the identity E ...
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 ( § ) , ξερ ( Τ ) , for every Borel set σ . If = { σ1 , ... , σn } , π ' = { σ1 ' , ... , σ'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 ( § ) , ξερ ( Τ ) , for every Borel set σ . If = { σ1 , ... , σn } , π ' = { σ1 ' , ... , σ'n ...
Contents
SPECTRAL OPERATORS | 1924 |
Introduction | 1927 |
Relations Between a Spectral Operator and Its Scalar | 1950 |
Copyright | |
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A₁ adjoint operator algebra of projections Amer arbitrary B*-algebra B₁ Boolean algebra Borel sets boundary conditions bounded linear operator bounded operator closed operator Colojoară commuting compact complex numbers complex plane contains converges Corollary countably additive Definition dense differential operator disjoint Doklady Akad E-measurable eigenvalues elements equation equivalent exists Foias follows from Theorem formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis identity inequality integral invariant inverse L₁ Lebesgue Lemma Math matrix multiplicity norm operators in Hilbert perturbation polynomial PROOF proved quasi-nilpotent resolution restriction Russian S₁ satisfies scalar operator scalar type operator scalar type spectral Section semi-group sequence shows spectral measure spectral operator spectral theory spectrum strong operator topology subset subspace sufficiently type spectral operator unbounded unique vector weakly complete zero