Linear maps that preserve the commutant, double commutant or the lattice of invariant subspaces

Abstract

For a linear map φ on either the algebra of all bounded operators on a Banach space of the algebra of all linear transformations on a vector space, we prove φ preserves any of the following functions of the operator if and only if it is of the form φ(T) = αT + f(T) I for a scalar α and a linear functional f:(i) the commutant, (ii) the double commutant, (iii) the algebra generated by the operator, (iv) the lattice of invariant subspaces, (v) the lattice of hyperinvariant subspace. (vi) the set of reducing subspaces.

^*Research partially supported by an NSERC grant and a University of Victoria faculty research grant.

^*Research partially supported by an NSERC grant and a University of Victoria faculty research grant.

Notes

^*Research partially supported by an NSERC grant and a University of Victoria faculty research grant.