Subspace hypercyclicity

A bounded linear operator T on Hilbert space is subspace-hypercyclic for a subspace M if there exists a vector whose orbit under T intersects the subspace in a relatively dense set. We construct examples to show that subspace-hypercyclicity is interesting, including a nontrivial subspace-hypercyclic operator that is not hypercyclic. There is a Kitai-like criterion that implies subspace-hypercyclicity and although the spectrum of a subspace-hypercyclic operator must intersect the unit circle, not every component of the spectrum will do so. We show that, like hypercyclicity, subspace-hypercyclicity is a strictly infinite-dimensional phenomenon. Additionally, compact or hyponormal operators can never be subspace-hypercyclic.Comment: 15 page

Subspace Polynomials and Cyclic Subspace Codes

Subspace codes have received an increasing interest recently due to their application in error-correction for random network coding. In particular, cyclic subspace codes are possible candidates for large codes with efficient encoding and decoding algorithms. In this paper we consider such cyclic codes and provide constructions of optimal codes for which their codewords do not have full orbits. We further introduce a new way to represent subspace codes by a class of polynomials called subspace polynomials. We present some constructions of such codes which are cyclic and analyze their parameters

Limit T-subspaces and the central polynomials in n variables of the Grassmann algebra

Let F be the free unitary associative algebra over a field F on the set X = {x_1, x_2, ...}. A vector subspace V of F is called a T-subspace (or a T-space) if V is closed under all endomorphisms of F. A T-subspace V in F is limit if every larger T-subspace W \gneqq V is finitely generated (as a T-subspace) but V itself is not. Recently Brand\~ao Jr., Koshlukov, Krasilnikov and Silva have proved that over an infinite field F of characteristic p>2 the T-subspace C(G) of the central polynomials of the infinite dimensional Grassmann algebra G is a limit T-subspace. They conjectured that this limit T-subspace in F is unique, that is, there are no limit T-subspaces in F other than C(G). In the present article we prove that this is not the case. We construct infinitely many limit T-subspaces R_k (k \ge 1) in the algebra F over an infinite field F of characteristic p>2. For each k \ge 1, the limit T-subspace R_k arises from the central polynomials in 2k variables of the Grassmann algebra G.Comment: 22 page

Robust subspace recovery by Tyler's M-estimator

This paper considers the problem of robust subspace recovery: given a set of $N$ points in $\mathbb{R}^D$, if many lie in a $d$-dimensional subspace, then can we recover the underlying subspace? We show that Tyler's M-estimator can be used to recover the underlying subspace, if the percentage of the inliers is larger than $d/D$ and the data points lie in general position. Empirically, Tyler's M-estimator compares favorably with other convex subspace recovery algorithms in both simulations and experiments on real data sets