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 2150
... contained in a spectral set of arbitrarily small diameter and thus in an ( T ) set of arbitrarily small diameter . Since it is clear that every subset of the resolvent set is an ( T ) set , condition ( D ) is im- mediate . let To verify ...
... contained in a spectral set of arbitrarily small diameter and thus in an ( T ) set of arbitrarily small diameter . Since it is clear that every subset of the resolvent set is an ( T ) set , condition ( D ) is im- mediate . let To verify ...
Page 2234
... contained in U , it may be supposed , since f ( T ) is independent of { e } , that ee1 . If x is in E ( e ) , then , by the paragraph preceding Definition 8 , f ( T | E ( en ) X ) x = f ( T | E ( e ) X ) x for n≥1 so that ƒ ( T ) x ...
... contained in U , it may be supposed , since f ( T ) is independent of { e } , that ee1 . If x is in E ( e ) , then , by the paragraph preceding Definition 8 , f ( T | E ( en ) X ) x = f ( T | E ( e ) X ) x for n≥1 so that ƒ ( T ) x ...
Page 2256
... contained in an arbitrarily small compact subset σ of o ( T ) which is open in the relative topology of o ( T ) . It follows that the set 7 ( 0 ) = { zz - 1e o } is a compact | 1 subset of o ( R ) , open in the relative topology of o ...
... contained in an arbitrarily small compact subset σ of o ( T ) which is open in the relative topology of o ( T ) . It follows that the set 7 ( 0 ) = { zz - 1e o } is a compact | 1 subset of o ( R ) , open in the relative topology of o ...
Contents
SPECTRAL OPERATORS | 1924 |
The Resolvent of a Spectral Operator | 1935 |
An Operational Calculus for Bounded Spectral | 1941 |
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