Locally linearly dependent operators
by
Peter Šemrl
University of Ljubljana
Coauthors: Matej Brešar

This is a report on a joint work with M. Bresar [2].

Let U and V be vector spaces over a field F. Linear operators T₁, ... , T_n :U --> V are locally linearly dependent if vectors T₁ u, ... , T_n u are linearly dependent in V for every u in U. Some problems concerning generalized polynomial identities [1], algebraic reflexivity and linear interpolation [4], and many others can be reduced to the problem of the description of locally linearly dependent operators. The complete description is known only in the cases when n is small. We present some improvments of the known results with simpler proofs. We also study countable families of locally linearly dependent bounded operators acting on Banach spaces. Our methods give a short proof of Müller's extension [5] of Kaplansky's result on locally algebraic operators [3].

References

1. S. A. Amitsur, Generalized polynomial identities and pivotal monomials, Trans. Amer. Math. Soc. 114 (1965), 210-226.
2. M. Bresar and P. Semrl, On locally linearly dependent operators and derivations, Trans. Amer. Math. Soc., to appear
3. I. Kaplansky, Infinite Abelian groups, Ann Arbor, 1954.
4. D. R. Larson, Reflexivity, algebraic reflexivity and linear interpolation, Amer. J. Math. 110 (1988), 283-299.
5. W. Müller, Kaplansky's theorem and Banach PI-algebras, Pacific J. Math. 141 (1990), 355-361.

Date received: November 2, 1998