arXiv subjects

Stanislav Shkarin

Publications and source records attributed to Stanislav Shkarin.

39 records · Page 3Linked to original sources

arXiv2010

Hypercyclic operators on topological vector spaces

Bonet, Frerick, Peris and Wengenroth constructed a hypercyclic operator on the locally convex direct sum of countably many copies of the Banach space $\ell_1$. We extend this result. In particular, we show that there is a hypercyclic operator on the locally convex direct sum of a sequence $\{X_n\}_{n\in\N}$ of Fréchet spaces if and only if each $X_n$ is separable and there are infinitely many $n\in\N$ for which $X_n$ is infinite dimensional. Moreover, we characterize inductive limits of sequences of separable Banach spaces which support a hypercyclic operator.

math.FA↗

arXiv2010

The Golod-Shafarevich inequality for Hilbert series of quadratic algebras and the Anick conjecture

We study the question on whether the famous Golod-Shafarevich estimate, which gives a lower bound for the Hilbert series of a (noncommutative) algebra, is attained. This question was considered by Anick in his 1983 paper 'Generic algebras and CW-complexes', Princeton Univ. Press., where he proved that the estimate is attained for the number of quadratic relations $d \leq \frac{n^2}{4}$ and $d \geq \frac{n^2}{2}$, and conjectured that this is the case for any number of quadratic relations. The particular point where the number of relations is equal to $ \frac{n(n-1)}{2}$ was addressed by Vershik. He conjectured that a generic algebra with this number of relations is finite dimensional. We prove that over any infinite field, the Anick conjecture holds for $d \geq \frac{4(n^2+n)}{9}$ and arbitrary number of generators $n$, and confirm the Vershik conjecture over any field of characteristic 0. We give also a series of related asymptotic results.

math.RA↗

arXiv2009

Operators commuting with the Volterra operator are not weakly supercyclic

We prove that any bounded linear operator on $L_p[0,1]$ for $1\leq p<\infty$, commuting with the Volterra operator $V$, is not weakly supercyclic, which answers affirmatively a question raised by Léon-Saavedra and Piqueras-Lerena. It is achieved by providing an algebraic flavored condition on an operator which prevents it from being weakly supercyclic and is satisfied for any operator commuting with $V$.

math.DS↗