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 1956
... THEOREM . If T is of finite type , its residual spectrum is void and a point is in its point spectrum if and only if ... follows from Theorem 2 that λ is in the point spectrum of T. If E ( { } ) = 0 then it follows from Theorem 2 that IT ...
... THEOREM . If T is of finite type , its residual spectrum is void and a point is in its point spectrum if and only if ... follows from Theorem 2 that λ is in the point spectrum of T. If E ( { } ) = 0 then it follows from Theorem 2 that IT ...
Page 2194
... follows from Theorem 3. Let A1 be the algebra of all operators of the form ƒ ƒ ( λ ) E ( dλ ) where ƒ is E - essentially bounded on o ( S ) . It follows from Theorem 10 that A1 is a full algebra of scalar type spectral operators which ...
... follows from Theorem 3. Let A1 be the algebra of all operators of the form ƒ ƒ ( λ ) E ( dλ ) where ƒ is E - essentially bounded on o ( S ) . It follows from Theorem 10 that A1 is a full algebra of scalar type spectral operators which ...
Page 2243
... follows from Theorem XV.5.1 that f ( T | E ( en ) X ) = Sen f ( x ) F ( dλ ) , so that = ƒ ( T | E ( en ) X ) E ( en ) x = ƒ_ƒ ( \ ) E ( dX ) x , It follows by Theorem 11 that en f ( T | E ( en ) X ) E ( en ) x = ƒ2 ( T ) E ( en ) x . x ...
... follows from Theorem XV.5.1 that f ( T | E ( en ) X ) = Sen f ( x ) F ( dλ ) , so that = ƒ ( T | E ( en ) X ) E ( en ) x = ƒ_ƒ ( \ ) E ( dX ) x , It follows by Theorem 11 that en f ( T | E ( en ) X ) E ( en ) x = ƒ2 ( T ) E ( en ) x . x ...
Contents
SPECTRAL OPERATORS | 1924 |
Relations Between a Spectral Operator and Its Scalar | 1950 |
The Algebras and | 1967 |
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 linear operator bounded operator closed operator commuting compact complex numbers complex plane converges Corollary countably additive Definition denote dense differential operator Doklady Akad domain E₁ 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