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 2070
... norm ( 16 ) | ƒ│0 = \ ƒ | 1 + \\ ƒ || , fe Lo , where || f || is the norm in X. We let A。 consist of all operators in which have the form ( 17 ) a = xe + f , fe Lo . Theorem 1 shows that the operator a determines a and ƒ uniquely ...
... norm ( 16 ) | ƒ│0 = \ ƒ | 1 + \\ ƒ || , fe Lo , where || f || is the norm in X. We let A。 consist of all operators in which have the form ( 17 ) a = xe + f , fe Lo . Theorem 1 shows that the operator a determines a and ƒ uniquely ...
Page 2450
... norm of an element A € 2 is simply the Hilbert - Schmidt norm of the operator RA . - = - If A , A and { A } is a Cauchy sequence , then , since R - 1 is bounded , | ( An - Am ) x | ≤ || A , — Am || ≤ | R - 1 | || R ( A , — A ...
... norm of an element A € 2 is simply the Hilbert - Schmidt norm of the operator RA . - = - If A , A and { A } is a Cauchy sequence , then , since R - 1 is bounded , | ( An - Am ) x | ≤ || A , — Am || ≤ | R - 1 | || R ( A , — A ...
Page 2462
... norm , and CT converges to zero in trace norm . PROOF . The set K = C ( { xe || x | ≤ 1 } ) is conditionally compact , and of elements of K thus for each ɛ > 0 there exists a finite set x1 , - .... n Хт such that each xe K satisfies ...
... norm , and CT converges to zero in trace norm . PROOF . The set K = C ( { xe || x | ≤ 1 } ) is conditionally compact , and of elements of K thus for each ɛ > 0 there exists a finite set x1 , - .... n Хт such that each xe K satisfies ...
Contents
SPECTRAL OPERATORS | 1924 |
The Resolvent of a Spectral Operator | 1935 |
An Operational Calculus for Bounded Spectral | 1941 |
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