Abstract:

Xd Bessel sequences, X_d frames, X_d independent frames, X_d tight frames and X_d Riesz basis for a Banach space X are introduced and discussed. It is proved that (B^Xd_X,‖·‖) is a Banach space when X_d is a BK-space. By defining an operator Tf, an isometric isomorphism from B^Xd_X to B(X*,X_d) is established when X_d is a BK-space and X is reflexive, which provides a necessary theoretical basis for studying Xd Bessel sequences by the operator theory. Finally, the equivalent characterizations of  X_d Bessel sequences for a Banach space X are given. Also, it is proved that independent X_d frames and X_d Riesz bases for a Banach space X are the same.

Key words:  X_d Bessel sequence; X_d frame; X_d Riesz basis