On the normalizer of the reflexive cover of a unital algebra of linear transformations

Abstract

Given a unital algebra A of linear transformations on a finite-dimensional complex vector space V, in this paper we study the set Col(A) consisting of those invertible linear transformations S on V which map every subspace M∈Lat(A) to a subspace SM∈Lat(A). We show that Col(A) is the normalizer of the group of invertible linear transformations in the reflexive cover of A. For the unital algebra (A) which is generated by a linear transformation A, we give the complete description of Col(A). By using the primary decomposition of A, we first reduce the problem of characterizing Col(A) to the problem of characterizing the group Col(N) of a given nilpotent linear transformation N. While Col(N) always contains all invertible linear transformations of the commutant (N)′ of N, it is always contained in the reflexive cover of (N)′. We prove that Col(N) is a proper subgroup of (AlgLat(N)′)−1 if and only if at least two Jordan blocks in the Jordan decomposition of N are of dimension 2 or more. We also determine the group Col(J2⊕J2).