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 |
Introduction | 1927 |
Relations Between a Spectral Operator and Its Scalar | 1950 |
Copyright | |
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A₁ adjoint operator algebra of projections Amer arbitrary B*-algebra B₁ Boolean algebra Borel sets boundary conditions bounded linear operator bounded operator closed operator Colojoară commuting compact complex numbers complex plane contains converges Corollary countably additive Definition dense differential operator disjoint Doklady Akad E-measurable eigenvalues elements equation equivalent exists Foias follows from Theorem formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis identity inequality integral invariant inverse L₁ Lebesgue Lemma Math matrix multiplicity norm operators in Hilbert perturbation polynomial PROOF proved quasi-nilpotent resolution restriction Russian S₁ satisfies scalar operator scalar type operator scalar type spectral Section semi-group sequence shows spectral measure spectral operator spectral theory spectrum strong operator topology subset subspace sufficiently type spectral operator unbounded unique vector weakly complete zero