More on positive commutators

Let A and B be positive operators on a Banach lattice E such that the commutator C=AB−BA is also positive. The paper continues the investigation of the spectral properties of C initiated in J. Bra?i? et al. (in press) [3]. If the sum A+B is a Riesz operator and the commutator C is a power compact operator, then C is a quasi-nilpotent operator having a triangularizing chain of closed ideals of E. If we assume that the operator A is compact and the commutator AC−CA is positive, the operator C is quasi-nilpotent as well. We also show that the commutator C is not invertible provided the resolvent set of C is connected.     

Spectral radius inequalities for positive commutators

We establish several inequalities for the spectral radius of a positive commutator of positive operators in a Banach space ordered by a normal and generating cone. The main purpose of this paper is to show that in order to prove the quasi-nilpotency of the commutator we do not have to impose any compactness condition on the operators under consideration. In this way we give a partial answer to the open problem posed in the paper by J. Bra?i?, R. Drnov?ek, Y. B. Farforovskaya, E. L. Rabkin, J. Zemánek (2010). Inequalities involving an arbitrary commutator and a generalized commutator are also discussed.     

Groups with a positive law on commutators

Let p be a prime number and let G be a finitely generated group that is residually a finite p-group. We prove that if G satisfies a positive law on all elements of the form [a,b][c,d]ⁱ, a,b,c,d∈G and i?0, then the entire derived subgroup G^′ satisfies a positive law. In fact, G^′ is an extension of a nilpotent group by a locally finite group of finite exponent.     

Norm inequalities for commutators of positive operators and applications

Let X, Y, and Z be operators on a Hilbert space such that X and Z are positive. It is shown that

On commutators of equivariant homeomorphisms

The long-standing problem of the perfectness of the compactly supported equivariant homeomorphism group on a G-manifold (with one orbit type) is solved in the affirmative. The proof is based on an argument different than that for the case of diffeomorphisms. The theorem is a starting point for computing H₁(HG(M)) for more complicated G-manifolds.     

On commutators of fractional integrals

Let be the infinitesimal generator of an analytic semigroup on with Gaussian kernel bounds, and let be the fractional integrals of for . For a BMO function on , we show boundedness of the commutators from to , where . Our result of this boundedness still holds when is replaced by a Lipschitz domain of with infinite measure. We give applications to large classes of differential operators such as the magnetic Schrödinger operators and second-order elliptic operators of divergence form.

     

A characterization of maximum norms of commutators of positive contractions

In this note, we give a characterization of a pair (A,B) of positive contractions with commutator AB−BA of maximum possible norm. A necessary and sufficient condition is that either or its complex conjugate is in the closure of the numerical range of AB.     

On generalized commutators and nil commutator ideals

Archiv der Mathematik -     

On groups with commutators of bounded order

Let be a prime, a non-negative integer. We prove that if is a residually finite group such that for all , then is locally finite.

     

On a certain type of commutators of operators

LetH be a separable infinite-dimensional Hilbert space and letC be a normal operator andG a compact operator onH. It is proved that the following four conditions are equivalent.

1. C +G is a commutatorAB-BA with self-adjointA.
2. There exists an infinite orthonormal sequencee _j inH such that |Σ _j ⁿ =1 (Ce _j, e_j)| is bounded.
3. C is not of the formC ₁ ⊕C ₂ whereC ₁ has finite dimensional domain andC ₂ satisfies inf {|(C ₂ x, x)|: ‖x‖=1}>0.
4. 0 is in the convex hull of the set of limit points of spC.

     

On the length of commutators in Chevalley groups

Let G = G(Φ,R) be the simply connected Chevalley group with root system Φ over a ring R. Denote by E(Φ,R) its elementary subgroup. The main result of the article asserts that the set of commutators C = {[a, b]|a ∈ G(Φ, R), b ∈ E(Φ, R)} has bounded width with respect to elementary generators. More precisely, there exists a constant L depending on Φ and dimension of maximal spectrum of R such that any element from C is a product of at most L elementary root unipotent elements. A similar result for Φ = A _l , with a better bound, was earlier obtained by Sivatski and Stepanov.     

On division rings with algebraic commutators of bounded degree

We shall answer several questions concerning additive or multiplicative commutators in a division ring which are algebraic of bounded degree over its center.     

On the annihilator of commutators with derivation in prime rings

SianoR un anello primo di caratteristica differente da 2,d una derivazione non nulla diR, L un ideale di Lie non centrale diR, a ∈ R. Sea[d(u), u] = 0, per ogni scelta diu ∈ L, alloraa = 0.