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 2325
... formulas ( 16 ) and ( 14 ) . It also follows , from Lemma 3.5 , formula ( 16 ) , and formula ( 14 ) , that the zero En of M ( u ) in R has the asymptotic representation = - - 00 §n ~ 2πn + x + z 2 a n− TM , + Σ 5mn - m , m = 1 and that ...
... formulas ( 16 ) and ( 14 ) . It also follows , from Lemma 3.5 , formula ( 16 ) , and formula ( 14 ) , that the zero En of M ( u ) in R has the asymptotic representation = - - 00 §n ~ 2πn + x + z 2 a n− TM , + Σ 5mn - m , m = 1 and that ...
Page 2334
... formula ( 26 ) and from the formula ( 21 ) giving the form of the kernel defining the operator G1 that if we excise from the angle A of the μ - plane ( defined by formula ( 8 ) ) a circle of radius & about each of the roots 27m + 2πс1 ...
... formula ( 26 ) and from the formula ( 21 ) giving the form of the kernel defining the operator G1 that if we excise from the angle A of the μ - plane ( defined by formula ( 8 ) ) a circle of radius & about each of the roots 27m + 2πс1 ...
Page 2340
... formula ( 42 ) . It now follows just as in Case 1A that the projection Em = E ( m ; T ) is given asymptotically by the formula ( 57 ) ( Emf ) ( x ) ~ n 2πί S Co 0 n Ĉ Axs ( μl + 2πm ) e1a + Aks ( μ + 2πm ) } ( k , j = 1 А ( μ + 2πm ) e1 ...
... formula ( 42 ) . It now follows just as in Case 1A that the projection Em = E ( m ; T ) is given asymptotically by the formula ( 57 ) ( Emf ) ( x ) ~ n 2πί S Co 0 n Ĉ Axs ( μl + 2πm ) e1a + Aks ( μ + 2πm ) } ( k , j = 1 А ( μ + 2πm ) e1 ...
Contents
SPECTRAL OPERATORS | 1924 |
The Resolvent of a Spectral Operator | 1935 |
An Operational Calculus for Bounded Spectral | 1941 |
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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