Linear Operators: Spectral operators |
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Page 2300
... finite dimensional range for all p . Thus , by Lemma VII.6.7 , I — E , has finite dimensional range for all sufficiently large p . Since E is a count- ably additive spectral resolution , we have E ( μ ; T + P ) ( I − E , ) = 0 if μ is ...
... finite dimensional range for all p . Thus , by Lemma VII.6.7 , I — E , has finite dimensional range for all sufficiently large p . Since E is a count- ably additive spectral resolution , we have E ( μ ; T + P ) ( I − E , ) = 0 if μ is ...
Page 2318
... finite dimensional range for all sufficiently large m , and hence , a fortiori , E has a finite dimensional range . Q.E.D. ∞0 11 THEOREM . Let T be the unbounded operator in Hilbert space defined by the formal differential operator ...
... finite dimensional range for all sufficiently large m , and hence , a fortiori , E has a finite dimensional range . Q.E.D. ∞0 11 THEOREM . Let T be the unbounded operator in Hilbert space defined by the formal differential operator ...
Page 2487
... finite dimensional complement , such that W ( H + T ) f = HWƒ for all ƒ in the domain of H. ( Hint : Use Exercise 14 , and induction on the dimension of the range of T. ) ( b ) Show that ac ( H + T ) has a finite dimensional complement ...
... finite dimensional complement , such that W ( H + T ) f = HWƒ for all ƒ in the domain of H. ( Hint : Use Exercise 14 , and induction on the dimension of the range of T. ) ( b ) Show that ac ( H + T ) has a finite dimensional complement ...
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