Vol. 55, No. 2, 1974

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Linear transformations on symmetric spaces

M. H. Lim

Vol. 55 (1974), No. 2, 499–505
Abstract

Let U be an n-dimensional vector space over an algebraically closed field F of characteristic zero, and let rU denote the r-th symmetric product space of U. Let T be a linear transformation on rU which sends nonzero decomposable elements to nonzero decomposable elements. We prove the following:

  1. If n = r + 1 then T is induced by a nonsingular transformation on T.
  2. If 2 < n < r + 1 then either T is induced by a nonsingular transformation on U or T(rU) = rW for some two dimensional subspace W of U.

The result for n > r+1 was recently obtained by L. J. Cummings.

Mathematical Subject Classification 2000
Primary: 15A69
Milestones
Received: 3 October 1973
Revised: 18 September 1974
Published: 1 December 1974
Authors
M. H. Lim