Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |
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Page 1934
... seen that x * x ( ) = 0 for all έ and all * II.3.14 , x ( § ) = 0 and thus x = ( §I − T ) x ( § ) = 0 . - X * . Hence , by Corollary 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 * II.3.14 , x ( § ) = 0 and thus x = ( §I − T ) x ( § ) = 0 . - X * . Hence , by Corollary 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 ( S ) E ( 0 ) = E ( 0 ) F ( § ) , ξερ ( Τ ) , = .... for every Borel set σ . If π = = { 01 , ... , σn } , π ' = { 01. o ...
... seen from Corollary XV.3.7 that F ( $ ) also commutes with the projec- tions in the range of E , that is , ( iii ) F ( S ) E ( 0 ) = E ( 0 ) F ( § ) , ξερ ( Τ ) , = .... for every Borel set σ . If π = = { 01 , ... , σn } , π ' = { 01. o ...
Page 2183
... seen that PA ( 7 ) = 2 ( B ) . Moreover , it is clear that ( IP ) B = R , and since R is closed , it will follow that ( I - P ) A ( 7 ) = R. i Thus , to show that A ( 7 ) = A ( B ) ✈R , it is sufficient to show that P is bounded on B ...
... seen that PA ( 7 ) = 2 ( B ) . Moreover , it is clear that ( IP ) B = R , and since R is closed , it will follow that ( I - P ) A ( 7 ) = R. i Thus , to show that A ( 7 ) = A ( B ) ✈R , it is sufficient to show that P is bounded on B ...
Contents
SPECTRAL OPERATORS | 1924 |
The Spectrum of a Spectral Operator | 1955 |
The Algebras A and | 1966 |
Copyright | |
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A₁ Acad adjoint operator algebra of projections Amer analytic arbitrary B-algebra B-space B*-algebra B₁ Banach space Boolean algebra Borel sets boundary conditions bounded linear operator bounded operator closed operator commuting compact complex numbers complex plane converges Corollary countably additive Definition dense differential operator Dokl Doklady Akad eigenvalues elements equation exists formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis ibid identity inequality invariant subspaces inverse Krein L₁ Lemma locally convex spaces multiplicity Nauk SSSR nonselfadjoint norm normal operators operators in Hilbert perturbation Proc PROOF properties prove Pures Appl quasi-nilpotent resolution Russian satisfies scalar operator scalar type operator scalar type spectral Section semi-group sequence shows spectral measure spectral operator spectral theory spectrum subset Suppose topology trace class type spectral operator unbounded uniformly bounded vector zero