Let ▫$A$▫ and ▫$B$▫ be ▫$n \times n$▫ real matrices with ▫$A$▫ symmetric and ▫$B$▫ skewsymmetric. Obviously, every simultaneously neutral subspace for the pair ▫$(A,B)$▫ is neutral for each Hermitian matrix ▫$X$▫ of the form ▫$X = \mu A + i\lambda B$▫, where ▫$\mu$▫ and ▫$\lambda$▫ are arbitrary real numbers. It is well-known that the dimension of each neutral subspace of ▫$X$▫ is at most ▫$\text{In}_+(X) + \text{In}_0(X)$▫, and similarly, the dimension of each neutral subspace of ▫$X$▫ is at most ▫$\text{In}_-(X) + \text{In}_0(X)$▫. These simple observations yield that the maximal possible dimension of an ▫$(A,B)$▫-neutral subspace is no larger than ▫$$\min \{ \min \{ \text{In}_+(\mu A+ i\lambda B) + \text{In}_0(\mu A + i\lambda B), \text{In}_-(\mu A + i\lambdaB) + \text{In}_0(\mu A + i\lambda B)\}\},$$▫ where the outer minimum is taken over all pairs of real numbers ▫$(\lambda, \mu)$▫. In this paper, it is proven that the maximal possible dimension of an ▫$(A,B)$▫-neutral subspace actually coincides with the above expression.
COBISS.SI-ID: 16067929
We show that if the quotient of a group by its absolute centre is locally finite of exponent ▫$n$▫, then the exponent of its autocommutator subgroup is ▫$n$▫-bounded, that is, bounded by a function depending only on ▫$n$▫. If the group itself is locally finite, then its exponent is ▫$n$▫-bounded as well. Under some extra assumptions, the exponent of its automorphism group is ▫$n$▫-bounded. We determine the absolute centre and autocommutator subgroup for a large class of (infinite) abelian groups.
COBISS.SI-ID: 15996761
This paper concerns analytic free maps. These maps are free analogs of classical analytic functions in several complex variables, and are defined in terms of non-commuting variables amongst which there are no relations - they are free variables. Analytic free maps include vector-valued polynomials in free (non-commuting) variables and form a canonical class of mappings fromvone non-commutative domain ▫$\mathcal{D}$▫ in say ▫$g$▫ variables to another non-commutative domain ▫$\tilde{\mathcal{D}}$▫ in ▫$\tilde{g}$▫ variables. As a natural extension of the usual notion, an analytic free map is proper if it maps the boundary of ▫$\mathcal{D}$▫ into the boundary of ▫$\tilde{\mathcal{D}}$▫. Assuming that both domains contain 0, we show that if ▫$f \colon \mathcal{D} \to \tilde{\mathcal{D}}$▫ is a proper analytic free map, and ▫$f(0)=0$▫, then ▫$f$▫ is one-to-one. Moreover, if also ▫$g = \tilde{g}$▫, then $f$ is invertible and ▫$f^{-1}$▫ is also an analytic free map. These conclusions on the map ▫$f$▫ are the strongest possible without additional assumptions on the domains ▫$\mathcal{D}$▫ and ▫$\tilde{\mathcal{D}}$▫.
COBISS.SI-ID: 15866201