Abstract
Since a frame for a Hilbert space must be a Bessel sequence, many results on (quasi-)affine bi-frame are established under the premise that the corresponding (quasi-)affine systems are Bessel sequences. However, it is very technical to construct a (quasi-)affine Bessel sequence. Motivated by this observation, in this paper we introduce the notion of weak (quasi-)affine bi-frame (W(Q)ABF) in a general reducing subspace for which the Bessel sequence hypothesis is not needed. We obtain a characterization of WABF, and prove the equivalence between WABF and WQABF under a mild condition. This characterization is used to recover some related known results in the literature.
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