Determinant preserving maps: an infinite dimensional version of a theorem of Frobenius

Nagy, Gergő (2016) Determinant preserving maps: an infinite dimensional version of a theorem of Frobenius. LINEAR AND MULTILINEAR ALGEBRA, inpres. inpress. ISSN 0308-1087

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Abstract

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 in nite 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 di er 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 semide nite matrices which preserve the determinant of linear combinations with xed coffcients.

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