Theorem ([11, Theorem 2, page 282]). Let R be a prime ring, L a noncommutative Lie ideal of R and d 6= 0 a derivation of R. If [d(x), x] ∈ Z(R), for all x ∈ L, then either R is commutative, or char(R) = 2 and R satisfies s4, the standard identity in 4 variables. Here we will examine what happens in case [d(x), x]n ∈ Z(R), for any x ∈ L, a noncommutative Lie ideal of R and n ≥ 1 a fixed integer....

We prove that for any odd integer N and any integer n > 0, the Nth power of a product of n commutators in a nonabelian free group of countable infinite rank can be expressed as a product of squares of 2n+1 elements and, for all such odd N and integers n, there are commutators for which the number 2n+1 of squares is the minimum number such that the Nth power of its product can be written as a pr...

If 3C is a separable (complex) Hubert space, and A is a (bounded, linear) operator on 3C, then A is a commutator if there exist operators B and C on 3C such that 4 = BCCB. I t was shown by Wintner [8] and also by Wielandt [7] that no nonzero scalar multiple of the identity operator I on 3C is a commutator, and this was improved by Halmos [5] who showed that no operator of the form X / + C is a ...

The operators on `∞ which are commutators are those not of the form λI + S with λ 6= 0 and S strictly singular.

We show that elements of unital C-algebras without tracial states are finite sums of commutators. Moreover, the number of commutators involved is bounded, depending only on the given C-algebra.

The operators on Lp = Lp[0, 1], 1 ≤ p < ∞, which are not commutators are those of the form λI + S where λ , 0 and S belongs to the largest ideal in L(Lp). The proof involves new structural results for operators on Lp which are of independent interest.

Various measures for the accuracy of approximate eigenstates of semibounded self-adjoint operators H in quantum theory, derived, e.g., by some variational technique, are scrutinized. In particular, the matrix elements of the commutator of the operatorH and (suitably chosen) different operators with respect to degenerate approximate eigenstates of H obtained by the variational methods are propos...

Subnormal operators arise naturally in complex function theory, differential geometry, potential theory, and approximation theory, and their study has rich applications in many areas of applied sciences as well as in pure mathematics. We discuss here some research problems concerning the structure of such operators: subnormal operators with finite-rank self-commutator, connections with quadratu...