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 2448
... norm ||| A ||| , A € 2. Let M1 and M2 be real numbers greater than zero . Suppose that : 1 2 ( a ) a continuous linear mapping : A → B ( X ) , of norm at most M1 , is given ; ( b ) a continuous linear mapping ŋ : A → A , of norm at ...
... norm ||| A ||| , A € 2. Let M1 and M2 be real numbers greater than zero . Suppose that : 1 2 ( a ) a continuous linear mapping : A → B ( X ) , of norm at most M1 , is given ; ( b ) a continuous linear mapping ŋ : A → A , of norm at ...
Page 2450
... norm of an element = A e A 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 , | ( A „ — A „ ) x | ≤ || A „ — A „ || ≤ | R¬1 | || R ( A „ — A ...
... norm of an element = A e A 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 , | ( A „ — A „ ) x | ≤ || A „ — A „ || ≤ | R¬1 | || R ( A „ — A ...
Page 2462
... norm , and CT converges to zero in trace norm . .... PROOF . The set K = C ( { x = H || x | ≤ 1 } ) is conditionally compact , and thus for each ɛ > 0 there exists a finite set x1 , xm of elements of K such that each xe K satisfies | x ...
... norm , and CT converges to zero in trace norm . .... PROOF . The set K = C ( { x = H || x | ≤ 1 } ) is conditionally compact , and thus for each ɛ > 0 there exists a finite set x1 , xm of elements of K such that each xe K satisfies | x ...
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