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 2323
... Hypothesis 1 is satisfied , we may write -1 π1 ( μ ) = αpμ3 + Ap - 1 μ3¬1 + + αo , ap # 0 , ( 13 ) π2 ( μ ) = b2μ2 + 6p - 1μ3 - 1 + ... • + bo , bp # 0 , πз ( μ ) = Сpμ3 + Cp - 1μ31 + 1 ... + Co. The labor of verifying Hypothesis 1 may ...
... Hypothesis 1 is satisfied , we may write -1 π1 ( μ ) = αpμ3 + Ap - 1 μ3¬1 + + αo , ap # 0 , ( 13 ) π2 ( μ ) = b2μ2 + 6p - 1μ3 - 1 + ... • + bo , bp # 0 , πз ( μ ) = Сpμ3 + Cp - 1μ31 + 1 ... + Co. The labor of verifying Hypothesis 1 may ...
Page 2397
... Hypothesis ( i ) of Theorem XVIII.2.34 is satisfied by virtue of Corollaries 9 and 11. Hypothesis ( ii ) has been established and is given by Lemma 7 ( iv ) . It therefore only remains to establish hypothesis ( iii ) of Theorem XVIII ...
... Hypothesis ( i ) of Theorem XVIII.2.34 is satisfied by virtue of Corollaries 9 and 11. Hypothesis ( ii ) has been established and is given by Lemma 7 ( iv ) . It therefore only remains to establish hypothesis ( iii ) of Theorem XVIII ...
Page 2401
... hypothesis ( b ) , to ( 4 ) Using hypothesis ( c ) , we may write this last equation as ( 5 ) q ( B — þ ( B , A1 ) ) = q ( A1 ) . Now , by hypothesis , the map B → ↓ ( B , A1 ) of A → A has norm at most M2 ( M1 + M2 ) -1 . Thus , by ...
... hypothesis ( b ) , to ( 4 ) Using hypothesis ( c ) , we may write this last equation as ( 5 ) q ( B — þ ( B , A1 ) ) = q ( A1 ) . Now , by hypothesis , the map B → ↓ ( B , A1 ) of A → A has norm at most M2 ( M1 + M2 ) -1 . Thus , by ...
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