Linear transformations of tensor products preserving a fixed rank

Dragomir Z. Djokovic

Vol. 30 (1969), No. 2, 411–414

DOI: 10.2140/pjm.1969.30.411

Abstract

In this paper T is a linear transformation from a tensor product X⊗ Yinto U ⊗V , where X,Y,U,V are vector spaces over an infinite field F. The main result gives a characterization of surjective transformations T for which there is a positive integer k(k < dimU,k < dimV ) such that whenever z ∈ X ⊗ Y has rank k then also Tz ∈ U ⊗ V has rank k. It is shown that T = A ⊗ B or T = S ∘ (C ⊗ D) where A,B,C,D are appropriate linear isomorphisms and S is the canonical isomorphism of V ⊗ U onto U ⊗ V .

Mathematical Subject Classification

Primary: 15.80

Milestones

Received: 21 August 1968

Published: 1 August 1969