Nilpotent elements of operator ideals as single commutators

Abstract

Abstract. For an arbitrary operator ideal I, every nilpotent element of I is a single commutator of operators from I t , for an exponent t that depends on the degree of nilpotency.

Five Hilbert Space Problems in Operator Algebras

“…In particular, if T ∈ C(B(H)) and π : B(H) → B(H)/K(H) is the quotient map, then π(T ) = α π(I ) for any α = 0, which implies that T / ∈ {α I + K : We stake no claim to authorship of this question, as it has been examined by a number of people in various specific C * -algebras and algebras of (Hilbert and Banach space) operators. For example, Dykema and Skripka have investigated commutators in type II 1 factors; Dykema, Figiel, Weiss and Wodzicki have done a detailed study of commutators in operator ideals [18]; and Dykema and Krishnaswamy-Usha [19] have studied nilpotent compact operators as commutators of compact operators. Recently, Dosev [10] and Dosev and Johnson [11] have classified commutators in B( 1 ) and in B( ∞ ) respectively (see also the work of Dosev et al [12]).…”

On commutators of compact operators via block tridiagonalization: generalizations and limitations of Anderson's approach

“…Then in [6, Section 7] the authors in 2004 obtained a sufficient condition in terms of ideals I, J in B(H) for an operator to be a single commutator of operators from I and J (we denote this class by [I, J ] 1 ) in the commutator ideal [I, J ]. And recently in [7], K. Dykema and A. Krishnaswamy-Usha proved that all nilpotent compact operators are commutators of compact operators. From this, focus began on trying to represent specific compact operators T and classes of compact operators as special kinds of commutators.…”

“…Later research mostly focused on commutator ideals in B(H), [I, J ], i.e., the finite linear span of commutators of operators from I with operators from J , in part because of the connection of [I, B(H)] to traces. (Chronologically, see [19,21,1,22,14,6,13,7].) The general characterization of commutator ideals [I, J ], was obtained in [6,Theorem 5.6] (or more simply distilled from [6, Introduction pp.…”