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 2296
... discrete operator . The space ( T ) is the set of all f in X for which ( T — XI ) - 1f is an entire function of λ . PROOF . If ( TAI ) -1f is entire , then by letting C be a small circle around λ , e σ ( T ) we find that 1 0 - 2πi ( TX ) ...
... discrete operator . The space ( T ) is the set of all f in X for which ( T — XI ) - 1f is an entire function of λ . PROOF . If ( TAI ) -1f is entire , then by letting C be a small circle around λ , e σ ( T ) we find that 1 0 - 2πi ( TX ) ...
Page 2361
... discrete and sp ( T + P ) = X ; - ( c ) if lim sup → ∞ μ ; < ∞ , and P ( T — λ 。 I ) ̄ ' is compact , then T + P is discrete and sp ( T + P ) = X. PROOF . By Theorem 6 and Lemma 5 , it suffices to show that in each of the cases ( a ) ...
... discrete and sp ( T + P ) = X ; - ( c ) if lim sup → ∞ μ ; < ∞ , and P ( T — λ 。 I ) ̄ ' is compact , then T + P is discrete and sp ( T + P ) = X. PROOF . By Theorem 6 and Lemma 5 , it suffices to show that in each of the cases ( a ) ...
Page 2362
... discrete and S∞ ( T + B ) = 0 . PROOF . This follows from Theorem 6 by placing v0 . Q.E.D. 9 COROLLARY . Let T be a discrete spectral operator in the reflexive B - space X. Suppose that all but a finite number of the points in o ( 1 ) ...
... discrete and S∞ ( T + B ) = 0 . PROOF . This follows from Theorem 6 by placing v0 . Q.E.D. 9 COROLLARY . Let T be a discrete spectral operator in the reflexive B - space X. Suppose that all but a finite number of the points in o ( 1 ) ...
Contents
SPECTRAL OPERATORS | 1924 |
The Resolvent of a Spectral Operator | 1935 |
An Operational Calculus for Bounded Spectral | 1941 |
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A₁ adjoint operator algebra of projections Amer analytic arbitrary B-algebra 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 denote dense differential operator Doklady Akad domain E₁ eigenvalues elements equation exists finite number follows from Lemma follows from Theorem formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis identity inequality integral invariant inverse L₁ Lebesgue Lemma Math multiplicity Nauk SSSR norm operators in Hilbert perturbation polynomial PROOF properties prove quasi-nilpotent resolution Russian S₁ satisfies scalar type operator scalar type spectral Section sequence shows spectral measure spectral operator spectral theory spectrum subset subspace Suppose trace class type spectral operator unbounded uniformly bounded unique vector zero