Some Properties of Canonical Dual K-Bessel Sequences for Parseval K-Frames. (29th July 2019) - Xiang, Zhong-Qi
Abstract : The concept of canonical dualK -Bessel sequences was recently introduced, a deep study of which is helpful in further developing and enriching the duality theory ofK -frames. In this paper we pay attention to investigating the structure of the canonical dualK -Bessel sequence of a ParsevalK -frame and some derived properties. We present the exact form of the canonical dualK -Bessel sequence of a ParsevalK -frame, and a necessary and sufficient condition for a dualK -Bessel sequence of a given ParsevalK -frame to be the canonical dualK -Bessel sequence is investigated. We also give a necessary and sufficient condition for a ParsevalK -frame to have a unique dualK -Bessel sequence and equivalently characterize the condition under which the canonical dualK -Bessel sequence of a ParsevalK -frame admits a unique dualK ⁎ -Bessel sequence. Finally, we obtain a minimal norm property on expansion coefficients of elements in the range ofK resulting from the canonical dualK -Bessel sequence of a ParsevalK -frame.