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 fin for which ( TAI ) -1f is an entire function of X. X - PROOF . If ( T — XI ) -1f is entire , then by letting C be a small circle around λ , e σ ( T ) we find that 0 = 1 - 1 [ ( T ...
... discrete operator . The space ( T ) is the set of all fin for which ( TAI ) -1f is an entire function of X. X - PROOF . If ( T — XI ) -1f is entire , then by letting C be a small circle around λ , e σ ( T ) we find that 0 = 1 - 1 [ ( T ...
Page 2361
... discrete and sp ( T + P ) = X ; - ( c ) if lim supμ < ∞ , and P ( T - λo I ) is discrete and sp ( T + P ) = X . - V is compact , then T + P PROOF . By Theorem 6 and Lemma 5 , it suffices to show that in each of the cases ( a ) , ( b ) ...
... discrete and sp ( T + P ) = X ; - ( c ) if lim supμ < ∞ , and P ( T - λo I ) is discrete and sp ( T + P ) = X . - V is compact , then T + P PROOF . By Theorem 6 and Lemma 5 , it suffices to show that in each of the cases ( a ) , ( b ) ...
Page 2362
... discrete and ( T + B ) = 0 ; ( b ) if lim info di≥ K > 0 , then there is a number ε = ε ( K , T ) > 0 such that TB is discrete and S∞ ( T + B ) = 0 whenever | B | ≤ ε ; ( c ) if lim inf∞ di > 0 , and B is compact , then T + B is ...
... discrete and ( T + B ) = 0 ; ( b ) if lim info di≥ K > 0 , then there is a number ε = ε ( K , T ) > 0 such that TB is discrete and S∞ ( T + B ) = 0 whenever | B | ≤ ε ; ( c ) if lim inf∞ di > 0 , and B is compact , then T + B is ...
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