Determinant Preserving Maps: an Infinite Dimensional Version of a Theorem of Frobenius

In this paper we investigate the structure of maps on classes of Hilbert space operators leaving the determinant of linear combinations invariant. Our main result is an infinite dimensional version of the famous theorem of Frobenius about determinant preserving linear maps on matrix algebras. In that theorem of ours, we use the notion of (Fredholm) determinant of bounded Hilbert space operators which differ from the identity by an element of the trace class. The other result of the paper describes the structure of those transformations on sets of positive semidefinite matrices which preserve the determinant of linear combinations with fixed coefficients. The determinant of square matrices (or linear operators on a finite dimensional vector space) is one of the most basic notions in matrix theory which has several applications also in other areas of mathematics. In light of its fundamental role, it is not surprising that maps on sets of matrices preserving related quantities have been extensively studied in the field of preserver problems. Indeed, the statement which is generally regarded as the first result in that branch of mathematics also concerns such transformations. It is the famous theorem of Frobenius from 1897 which reads as follows. In this paper, for a positive integer n and a field F , the space of all n× n matrices with entries in F is denoted by Mn(F ) and t stands for the transpose. 2010 Mathematics Subject Classification. Primary: 47B49. Secondary: 47B10, 15B48.