Linear Operators: Spectral operators |
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Page 1930
Nelson Dunford, Jacob T. Schwartz. 2 DEFINITION . If Σ is a Boolean algebra of subsets of the complex plane which contains the void set and the whole plane , in short , if Σ is a field of sets in the complex plane , then a spectral measure ...
Nelson Dunford, Jacob T. Schwartz. 2 DEFINITION . If Σ is a Boolean algebra of subsets of the complex plane which contains the void set and the whole plane , in short , if Σ is a field of sets in the complex plane , then a spectral measure ...
Page 2145
... field ( T ) or the o - field M ( T ) may not contain all Borel sets . The first of these difficulties will be eliminated by the hypo- thesis ( C ) , to be made presently . The second of these difficulties ... OF THE CONDITIONS ( A , B , C )
... field ( T ) or the o - field M ( T ) may not contain all Borel sets . The first of these difficulties will be eliminated by the hypo- thesis ( C ) , to be made presently . The second of these difficulties ... OF THE CONDITIONS ( A , B , C )
Page 2148
... the family of all Borel sets . To complete the proof it will suffice to show that o ( T ) ≤ 8 for every Borel set 8. If λ 8 , then , using ( D ) , the compact set So ( T ) may be covered by a set σ in the field ( T ) with Aσ . Since T is a ...
... the family of all Borel sets . To complete the proof it will suffice to show that o ( T ) ≤ 8 for every Borel set 8. If λ 8 , then , using ( D ) , the compact set So ( T ) may be covered by a set σ in the field ( T ) with Aσ . Since T is a ...
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