Linear Operators: Spectral operators |
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Page 1926
... arbitrary T. To see more clearly the difference between the calculi given by these two formulas , let us rewrite them by introducing the nilpotent N and the resolution of the identity E defined by the equations T N = 1- Σ λε ( λ ) , E ...
... arbitrary T. To see more clearly the difference between the calculi given by these two formulas , let us rewrite them by introducing the nilpotent N and the resolution of the identity E defined by the equations T N = 1- Σ λε ( λ ) , E ...
Page 1941
... arbitrary , it follows that o ( N ) = { 0 } . It then follows from Corollary 3 that N is a quasi - nilpotent . Q.E.D. 6 DEFINITION . The decomposition , given in Theorem 5 , of a spectral operator TS + N into a sum of a scalar type ...
... arbitrary , it follows that o ( N ) = { 0 } . It then follows from Corollary 3 that N is a quasi - nilpotent . Q.E.D. 6 DEFINITION . The decomposition , given in Theorem 5 , of a spectral operator TS + N into a sum of a scalar type ...
Page 1963
... arbitrary , shows that by ay . Since B ( 5 " ) is a B * -algebra ( IX.3.2 ) under the involution . A → A * , it follows that the map ( a1 ; ) → ( b1 , ) is an involution in the algebra M , ( B ( S ) ) and that , with this involution ...
... arbitrary , shows that by ay . Since B ( 5 " ) is a B * -algebra ( IX.3.2 ) under the involution . A → A * , it follows that the map ( a1 ; ) → ( b1 , ) is an involution in the algebra M , ( B ( S ) ) and that , with this involution ...
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