Linear Operators, Part 2 |
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Page 2356
... discrete and ( T + P ) = 0 ; ( b ) if lim supμ ≤ K≤∞ , there exists a 88 ( K , T ) > 0 such that if | P ( TI ) | < 8 , then T + P is discrete and S∞ ( T + P ) = 0 ; ( c ) if lim sup∞ μ ; < ∞ , and P ( T — λ 。 I ) ̄ ” is compact ...
... discrete and ( T + P ) = 0 ; ( b ) if lim supμ ≤ K≤∞ , there exists a 88 ( K , T ) > 0 such that if | P ( TI ) | < 8 , then T + P is discrete and S∞ ( T + P ) = 0 ; ( c ) if lim sup∞ μ ; < ∞ , and P ( T — λ 。 I ) ̄ ” is compact ...
Page 2361
... discrete and sp ( T + P ) = X ; - V ( c ) if lim sup → ∞ μi < ∞ , 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 ; - V ( c ) if lim sup → ∞ μi < ∞ , 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 ( T + B ) = 0 . PROOF . This follows from Theorem 6 by placing v = 0 . 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 ( T + B ) = 0 . PROOF . This follows from Theorem 6 by placing v = 0 . 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 Canonical Reduction of a Spectral Operator | 1939 |
Bounded Spectral Operators in Hilbert Space | 1947 |
Copyright | |
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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 Borel function 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 eigenvalues elements equation exists finite number follows from Lemma 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