On commutator subgroups

Acta Mathematica Hungarica -     

On commutator subgroups of Artin groups

Doklady Mathematics -     

On the commutator of the Marcinkiewicz integral

boundedness is considered for the commutator of higher-dimensional Marcinkiewicz integral. Some conditions implying the and the boundedness for the commutator of the Marcinkiewicz integral are obtained.     

On the positive commutator in the radical

In this paper we prove that a positive commutator between a positive compact operator A and a positive operator B is in the radical of the Banach algebra generated by A and B. Furthermore, on every at least three-dimensional Banach lattice we construct finite rank operators A and B satisfying \(AB\ge BA\ge 0\) such that the commutator \(AB-BA\) is not contained in the radical of the Banach algebra generated by A and B. These two results now completely answer to two open questions published in (Bra?i? et al., Positivity 14:431–439, 2010). We also obtain relevant results in the case of the Volterra and the Donoghue operator.     

On generalized commutators and nil commutator ideals

Archiv der Mathematik -     

On some properties of the decoding of admissible words for primitive substitutions

A number of properties of decoding for admissible sequences are studied, some of which were studied earlier, and some are new. Several conditions under which a substitution possesses these properties are obtained. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 127–136. Supported by Russian Foundation for Basic Research, grant 94-01-00921. Translated by A. N. Livshits.     

On the commutator length of a Dehn twist

We show that on a nonorientable surface of genus at least 7 any power of a Dehn twist is equal to a single commutator in the mapping class group and the same is true, under additional assumptions, for the twist subgroup, and also for the extended mapping class group of an orientable surface of genus at least 3.     

The commutator width of some relatively free lie algebras and nilpotent groups

We determine the exact values of the commutator width of absolutely free and free solvable Lie rings of finite rank, as well as free and free solvable Lie algebras of finite rank over an arbitrary field. We calculate the values of the commutator width of free nilpotent and free metabelian nilpotent Lie algebras of rank 2 or of nilpotency class 2 over an arbitrary field. We also find the values of the commutator width for free nilpotent and free metabelian nilpotent Lie algebras of finite rank at least 3 over an arbitrary field in the case that the nilpotency class exceeds the rank at least by 2. In the case of free nilpotent and free metabelian nilpotent Lie rings of arbitrary finite rank, as well as free nilpotent and free metabelian nilpotent Lie algebras of arbitrary finite rank over the field of rationals, we calculate the values of commutator width without any restrictions. It follows in particular that the free or nonabelian free solvable Lie rings of distinct finite ranks, as well as the free or nonabelian free solvable Lie algebras of distinct finite ranks over an arbitrary field are not elementarily equivalent to each other. We also calculate the exact values of the commutator width of free ?-power nilpotent, free nilpotent, free metabelian, and free metabelian nilpotent groups of finite rank.     

Marcinkiewicz积分交换子在一些Hardy空间的有界性

讨论了由核函数满足具有某类Dini条件的Marcinkiewicz积分μΩ及函数b∈Lipβ(R~n)生成的交换子μ(_Ω,b)~m的性质.证明了Marcinkiewicz积分交换子μ_(_Ω,b)~m在Hardy型空间H_(bm,s)(R~n)上有界,也在Herz型Hardy空间H_(bm)K_p~(a_q)(R~n)上有界.     

On a commutator theorem of robert c. thompson

A very short proof of the following special case of R. C. Thompson's Theorem is presented. If A is an n × n mairix over a field with more than n elements and if det A = 1. then Ais a multiplicative commutator.     

On the commutator subgroup of a nonconnected central group

In this paperG denotes a central topologicalT ₂-group—G/Z(G) compact, whereZ(G) is the center. There are some results concerning compactness of the commutator subgroupG; in general (G)^– is compact ([3]), but not necessarilyG ([7]). If in additionG is a Lie group or ifG is connected,G is compact ([6], [5]). The purpose of this paper is to show, that if the componentG ₀ of the identity is open,G must be compact, and to give an example of a compact group with (G/G ₀) compact, whileG is not compact.Dedicated to Prof. R. Inzinger on his 70th birthday     

On a commutator theorem of robert c. thompson

A very short proof of the following special case of R. C. Thompson's Theorem is presented. If A is an n × n mairix over a field with more than n elements and if det A = 1. then Ais a multiplicative commutator.     

On a variant of commutator estimates in spectral theory

Completeness is proved for systems of two or three quantum particles. The proof is based on the following statement on operators in a Hilbert space. If an operator A is bounded with respect to a self-adjoint operator H and Re((H – )f, Af) Bf², then B is smooth with respect to H.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 29–36, 1987.