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 2239
... follows from Lemma 6 that T ( f ) is a closed , densely defined operator . Moreover , statement ( g ) follows from Corollary 7 . Statement ( d ) is obvious . Letting e e Σ 。 and x = E ( e ) X , we have - T ( fxe ) x : = lim T ( fxe ) E ...
... follows from Lemma 6 that T ( f ) is a closed , densely defined operator . Moreover , statement ( g ) follows from Corollary 7 . Statement ( d ) is obvious . Letting e e Σ 。 and x = E ( e ) X , we have - T ( fxe ) x : = lim T ( fxe ) E ...
Page 2246
... follows that R ( A ) is a bounded operator whose range is contained in the domain of C. It is clear then that ( AI —C ) R ( \ ) x = x for x in H and R ( A ) ( XIC ) x = x for x in D ( C ) , so that R ( A ) = R ( A ; C ) and λ σ ( C ) ...
... follows that R ( A ) is a bounded operator whose range is contained in the domain of C. It is clear then that ( AI —C ) R ( \ ) x = x for x in H and R ( A ) ( XIC ) x = x for x in D ( C ) , so that R ( A ) = R ( A ; C ) and λ σ ( C ) ...
Page 2459
... follows at once . - If an ac ( H ) and lim , xx , then , by what we have already proved , we may write x Y1 + Y2 + Y3 , where y1 = [ ac ( H ) and y2 , Y3 are orthogonal to Zac ( H ) . But , since x , ac ( H ) we have ( x , y ) = 0 for ...
... follows at once . - If an ac ( H ) and lim , xx , then , by what we have already proved , we may write x Y1 + Y2 + Y3 , where y1 = [ ac ( H ) and y2 , Y3 are orthogonal to Zac ( H ) . But , since x , ac ( H ) we have ( x , y ) = 0 for ...
Contents
SPECTRAL OPERATORS | 1924 |
An Operational Calculus for Bounded Spectral | 1941 |
Bounded Spectral Operators in Hilbert Space | 1947 |
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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 inverse Krein L₁ Lemma locally convex spaces multiplicity Nauk SSSR norm normal operators operators in Hilbert perturbation polynomial Proc PROOF properties prove Pure Appl quasi-nilpotent resolution Russian S₁ 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