Linear Operators: Spectral operators |
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Page 2296
... discrete operator . The space 6 all f in X for which ( T — XI ) -1f is an entire function of λ . ( T ) is the set of PROOF . If ( TAI ) -1f is entire , then by letting C be a small circle around λ , e σ ( T ) we find that 0 - 1 - 2 ...
... discrete operator . The space 6 all f in X for which ( T — XI ) -1f is an entire function of λ . ( T ) is the set of PROOF . If ( TAI ) -1f is entire , then by letting C be a small circle around λ , e σ ( T ) we find that 0 - 1 - 2 ...
Page 2356
... discrete and ( T + P ) = 0 ; ( b ) if lim sup → ∞ μi ≤ k ≤∞ , there exists a 8 = S ( K , T ) > 0 such that if | P ( T — λ 。 I ) ̄ ' | < 8 , then T + P is discrete and S∞ ( T + P ) = 0 ; ( c ) if lim sup∞0 μ , < ∞ , and P ( T ...
... discrete and ( T + P ) = 0 ; ( b ) if lim sup → ∞ μi ≤ k ≤∞ , there exists a 8 = S ( K , T ) > 0 such that if | P ( T — λ 。 I ) ̄ ' | < 8 , then T + P is discrete and S∞ ( T + P ) = 0 ; ( c ) if lim sup∞0 μ , < ∞ , and P ( T ...
Page 2362
... discrete and ( TB ) = 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 ( T ...
... discrete and ( TB ) = 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 ( T ...
Contents
SPECTRAL OPERATORS | 1924 |
Spectral Operators | 1925 |
Terminology and Preliminary Notions | 1928 |
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 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