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 1994
... converges , we see from the continuity of F that lim ƒ | Ã ‚ ( 8 ) — Xm ( s ) | 2 ds = 0 , m.n ― ∞ P - which implies ( III.3.6 ) that X , converges in measure on σ . Conversely , if ( 29 ) and ( 30 ) hold , then , since the norms | A ...
... converges , we see from the continuity of F that lim ƒ | Ã ‚ ( 8 ) — Xm ( s ) | 2 ds = 0 , m.n ― ∞ P - which implies ( III.3.6 ) that X , converges in measure on σ . Conversely , if ( 29 ) and ( 30 ) hold , then , since the norms | A ...
Page 2218
... converges weakly to a projection , then it converges strongly . = PROOF . In view of Lemma 23 , the proof may be restricted to the case where the Boolean algebra B is complete . Let { E } be a weakly convergent generalized sequence in B ...
... converges weakly to a projection , then it converges strongly . = PROOF . In view of Lemma 23 , the proof may be restricted to the case where the Boolean algebra B is complete . Let { E } be a weakly convergent generalized sequence in B ...
Page 2462
... converges uniformly to zero and { CT * } converges uniformly to zero . Moreover , if C belongs to the trace class C1 , then TC converges to zero in trace norm , and CT converges to zero in trace norm . - PROOF . The set KC ( { xe || x ...
... converges uniformly to zero and { CT * } converges uniformly to zero . Moreover , if C belongs to the trace class C1 , then TC converges to zero in trace norm , and CT converges to zero in trace norm . - PROOF . The set KC ( { xe || x ...
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