Vol. 23, No. 3, 1967

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Transformations on tensor spaces

Roy Westwick

Vol. 23 (1967), No. 3, 613–620

In this paper we consider those linear transformations from one tensor product of vector spaces to another which carry nonzero decomposable tensors into nonzero decomposable tensors. We obtain a general decomposition theorem for such transformations. If we suppose further that the transformation maps the space into itself then we have a complete structure theorem in the following two cases: (1) the transformation is onto, and (2) the field is algebraically closed and the tensor space is a product of finite dimensional vector spaces. The main results are contained in Theorems 3.5 and 3.8 which state that the transformation T : U1 Un U1 Un has the form T(x1 xn) = T1(xπ(1)) Tn(x2ζ(n)) where Ti : Uπ(i) Ui are nonsingular and π is a permutation. Case (2) generalizes a theorem of Marcus and Moyls.

Mathematical Subject Classification
Primary: 15.85
Received: 2 May 1966
Published: 1 December 1967
Roy Westwick