Bounded reflexivity of kernels of elementary operators

Abstract

Kernels of elementary operators of length two give rise to many interesting classes of operators. In this paper we study bounded reflexivity of these kernels. For general Banach spaces, a sufficient condition is given for the kernels to be boundedly reflexive, in particular this assures that the commutant of any Banach space operator is boundedly reflexive. We also show that the bounded reflexive cover of the set of all Toeplitz matrices in Mm×n is the entire Mm×n. For finite-dimensional Banach spaces, using Kronecker canonical form for matrix pencils, we obtain some necessary and sufficient conditions for the kernels to be boundedly reflexive. These conditions enable us to easily determine bounded reflexivity by simply computing rank or nullity of matrices.