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 2256
... Suppose that o ( T ) is totally disconnected . Then T is a spectral operator if and only if ( a ) the family of projections E ( o ; T ) corresponding to compact spectral sets of T is uniformly bounded , and ( b ) no non - zero x in X ...
... Suppose that o ( T ) is totally disconnected . Then T is a spectral operator if and only if ( a ) the family of projections E ( o ; T ) corresponding to compact spectral sets of T is uniformly bounded , and ( b ) no non - zero x in X ...
Page 2284
... suppose the sets en are disjoint and U - 1 en P , the whole plane . n = - n n Now consider n fixed and suppose E , 0. It follows from Lemma 18 and Theorem 19 that there exist vectors x1 , x , and functions x , n j * .... Xn x * such ...
... suppose the sets en are disjoint and U - 1 en P , the whole plane . n = - n n Now consider n fixed and suppose E , 0. It follows from Lemma 18 and Theorem 19 that there exist vectors x1 , x , and functions x , n j * .... Xn x * such ...
Page 2303
... Suppose that E is its resolution of the identity , and suppose that { λn } is an enumeration of its spectrum . Let d , denote the distance from λn to o ( T ) — { \ n } . Suppose that for all but a finite number of n , E ( λn ) has a one ...
... Suppose that E is its resolution of the identity , and suppose that { λn } is an enumeration of its spectrum . Let d , denote the distance from λn to o ( T ) — { \ n } . Suppose that for all but a finite number of n , E ( λn ) has a one ...
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