On Operator Commutators

Tohoku Mathematical Journal, 1975

2002

Let D be a derivation on a Banach algebra; by using the operator D 2,wegive necessary and sufficient conditions for the separating ideal of D to be nilpotent. We also introduce an ideal M(D) and apply it to find out more equivalent conditions for the continuity of D and for nilpotency of its separating ideal. 2000 Mathematics Subject Classification: 46H40, 47B47. 1. Introduction. Let A be a Banach algebra. By a derivation on A, wemean a linear mapping D: A → A, which satisfies D(ab) = aD(b) + D(a)b for all a and b in A. The separating space of D is the set S(D) = { a ∈ A: ∃ {} an ⊂ A; an � → 0, D ( )}

Journal of Functional Analysis, 1986

Let L(X) be the algebra of all bounded operators on a non-trivial complex Banach space X and F: L(X) + L(X) a bijective linear operator such that F and F-' both send commuting pairs of operators into commuting pairs. Then, either F(A)=aUAW'+p(A)I, or F(A)=uUA'W'+p(A) I, where p is a linear functional on L(X), U is a bounded linear bijective operator between the appropriate two spaces, cr is a complex constant, and A' is the adjoint of A. The form of an operator F for which F and F-' both send projections of rank one into projections of rank one is also determined. I(" 1986 Academic Press. Inc

1981

Integral Equations and Operator Theory

VX e s fA(X) = AX - XA; we denote R(~A), R(~A)- and {A}' respectively the range, the norm closure of the range and the kernel of ~A. We denote Af = {A e/:(g) : R(~A)- (2 {A*}' = {0}}. If H is finite-dimensional, Af = s If H is infinite-dimensional, this equality does not hold. So a reasonable purpose is to determine what elements are in N. When H is a separable Hilbert space, AY contains the operators A for which p(A) is normal for some quadratic polynomial p(z) [2 ],the subnormal operators with cyclic vectors [2 ] and the ison:letries [3 ]. In this paper, we show that AY contains also all the operators unitarity equivalent to Jordan operators.

Journal of Mathematical Analysis and Applications, 2005

We characterize the bounded linear operators T defined on Banach spaces satisfying a-Browder's theorem, or a-Weyl's theorem, by means of the discontinuity of some maps defined on certain subsets of C. Several other characterizations are given in terms of localized SVEP, as well as by means of the quasi-nilpotent part, the hyper-kernel or the analytic core of λI − T . 531 denote the class of all upper semi-Fredholm operators, and let Φ − (X) := T ∈ L(X): β(T ) < ∞ denote the class of all lower semi-Fredholm operators. The class of all semi-Fredholm operators is defined by

Archiv Der Mathematik, 1993

Journal of Algebra, 1986

1996

Archiv der Mathematik, 1995