Leo Livshits - Locally linearly dependent spaces of matrices

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Leo Livshits - Locally linearly dependent spaces of matrices

	LEO LIVSHITS, Colby College, Waterville, Maine 04901, USA
	Locally linearly dependent spaces of matrices

(Joint work with D. Hadwin and B. Mathes)

A subspace $\Psi$ of ${M}_{m\times n}({\bf C})$ is said to be locally linearly dependent (``LLD'') if $\{T(x)\mid T\in \Psi \}$ is linearly dependent for every $x \in {\bf F}^n$ . Locally linearly independent (i.e. not dependent) spaces are exactly those possessing a separating vector. If such a space is proper and does not have a one-dimensional $\textrm{LLD}$ extention, then it is reflexive. (Converse is false.)

We will present several results about $\textrm{LLD}$ spaces. For example: no $\textrm{LLD}$ subspace of $M_{n}({\bf C})$ , $n\geq 3$ can contain both an invertible and a cyclic operator.

eo@camel.math.ca