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 2141
... assume without loss of generality that σ is void . Since E ( om ) = E ( σm0 ( T ) ) , by the preceding lemma , we may also assume that σm ≤ o ( T ) . Suppose that our assertion is false , so that there is a p > 0 and a vector x such ...
... assume without loss of generality that σ is void . Since E ( om ) = E ( σm0 ( T ) ) , by the preceding lemma , we may also assume that σm ≤ o ( T ) . Suppose that our assertion is false , so that there is a p > 0 and a vector x such ...
Page 2150
... assumed , without loss of generality , that d≤ σ ( T ) . Since o ( T ) is totally discon- nected , the closed set 8 is an ... assume also 0 8 = that I is smoothly embedded in a 2150 XVI . SPECTRAL OPERATORS : SUFFICIENT CONDITIONS XVI.5.3.
... assumed , without loss of generality , that d≤ σ ( T ) . Since o ( T ) is totally discon- nected , the closed set 8 is an ... assume also 0 8 = that I is smoothly embedded in a 2150 XVI . SPECTRAL OPERATORS : SUFFICIENT CONDITIONS XVI.5.3.
Page 2212
... assume , in the proof of ( ix ) , that σr = σy . Since A ( xy ) = Ax- Ay it follows that - T ( ƒx − y ) ( x − y ) = T ( ƒ2 ) x — T ( fy ) Y , and consequently that = T ( fz - y - fx ) xT ( ƒz - v —ƒy ) y . To complete the proof of ...
... assume , in the proof of ( ix ) , that σr = σy . Since A ( xy ) = Ax- Ay it follows that - T ( ƒx − y ) ( x − y ) = T ( ƒ2 ) x — T ( fy ) Y , and consequently that = T ( fz - y - fx ) xT ( ƒz - v —ƒy ) y . To complete the proof of ...
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