Linear Operators: Spectral operators |
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Page 1994
... converges , we see from the continuity of F that = lim [ | Xn ( s ) — Xm ( s ) | 2 ds = 0 , ∞ ← u'w Φ σ which implies ( III.3.6 ) that X converges in measure on σ . Conversely , if ( 29 ) and ( 30 ) hold , then , since the norms | λ ...
... converges , we see from the continuity of F that = lim [ | Xn ( s ) — Xm ( s ) | 2 ds = 0 , ∞ ← u'w Φ σ which implies ( III.3.6 ) that X converges in measure on σ . Conversely , if ( 29 ) and ( 30 ) hold , then , since the norms | λ ...
Page 2218
... converges weakly to a projection , then it converges strongly . α 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 . α 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 2319
... converges to f unconditionally in the topology of An ) ( J ) . ∞ = PROOF . The series 1 E ( A ; T ) f certainly converges uncondi- tionally in the topology of L2 ( J ) . On the other hand , so does the series = T ( Σ E ( A ,; T ) S ) ...
... converges to f unconditionally in the topology of An ) ( J ) . ∞ = PROOF . The series 1 E ( A ; T ) f certainly converges uncondi- tionally in the topology of L2 ( J ) . On the other hand , so does the series = T ( Σ E ( A ,; T ) S ) ...
Contents
SPECTRAL OPERATORS | 1924 |
14 | 1983 |
Sufficient Conditions | 2134 |
Copyright | |
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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