Minimal and maximal unconditional bases with respect to framings.

This paper studies framings in Banach spaces, a concept raised by Casazza, Han and Larson, which is a natural generalization of traditional frames in Hilbert spaces and unconditional bases in Banach spaces. The minimal unconditional bases and the maximal unconditional bases with respect to framings are introduced. Our main result states that, if ( x_i, f_i) is a framing of a Banach space X, and ( e_i^min) and ( e_i^max) are the minimal unconditional basis and the maximal unconditional basis with respect to ( x_i, f_i), respectively, then for any unconditional basis ( e_i) associated with ( x_i, f_i), there are A,B > 0 such that [Figure not available: see fulltext.]. It means that for any framing, the corresponding associated unconditional bases have common upper and lower bounds. [ABSTRACT FROM AUTHOR]