Hilbert–Schmidt frames and their duals

The concept of Hilbert–Schmidt frame (HS-frame) was first introduced by Sadeghi and Arefijamaal in 2012. It is more general than [Formula: see text]-frames, and thus, covers many generalizations of frames. This paper addresses the theory of HS-frames. We present a parametric and algebraic formula for all duals of an arbitrarily given HS-frame; prove that the canonical HS-dual induces a minimal-norm expression of the elements in Hilbert spaces; characterize when an HS-frame is an HS-Riesz basis, and when an HS-Bessel sequence is an HS-Riesz sequence (HS-Riesz basis) in terms of Gram matrices.