Journal of the
Korean Mathematical Society JKMS

Two standard Bessel sequences in a Hilbert $C^\ast$-module are approximately duals if the distance (with respect to the norm) between the identity operator on the Hilbert $C^\ast$-module and the operator constructed by the composition of the synthesis and analysis operators of these Bessel sequences is strictly less than one. In this paper, we introduce $(a,m)$-approximate duality using the distance between the identity operator and the operator defined by multiplying the Bessel multiplier with symbol $m$ by an element $a$ in the center of the $C^\ast$-algebra. We show that approximate duals are special cases of $(a,m)$-approximate duals and we generalize some of the important results obtained for approximate duals to $(a,m)$-approximate duals. Especially we study perturbations of $(a,m)$-approximate duals and $(a,m)$-approximate duals of modular Riesz bases.

Keywords: Hilbert $C^\ast$-module, Bessel multiplier, approximate duality, modular Riesz basis

MSC numbers: 42C15, 46H25, 47A05