Banach frames are defined by straightforward generalization of (Hilbert space) frames. We characterize Banach frames (and X d -frames) in separable Banach spaces, and relate them to series expansions in Banach spaces. In particular, our results show that we can not expect Banach frames to share all the nice properties of frames in Hilbert spaces.

In the present paper the unconditional convergence and the invertibility of multipliers is investigated. Multipliers are operators created by (framelike) analysis, multiplication by a fixed symbol, and resynthesis. Sufficient and/or necessary conditions for unconditional convergence and invertibility are determined depending on the properties of the analysis and synthesis sequences, as well as the symbol. Examples which show that the given assertions cover different classes of multipliers are given. If a multiplier is invertible, a formula for the inverse operator is determined. The case when one of the sequences is a Riesz basis is completely characterized.

Certain mathematical objects appear in a lot of scientific disciplines, like physics, signal processing and, naturally, mathematics. In a general setting they can be described as frame multipliers, consisting of analysis, multiplication by a fixed sequence (called the symbol), and synthesis. In this paper we show a surprising result about the inverse of such operators, if any, as well as new results about a core concept of frame theory, dual frames. We show that for semi-normalized symbols, the inverse of any invertible frame multiplier can always be represented as a frame multiplier with the reciprocal symbol and dual frames of the given ones. Furthermore, one of those dual frames is uniquely determined and the other one can be arbitrarily chosen. We investigate sufficient conditions for the special case, when both dual frames can be chosen to be the canonical duals. In connection to the above, we show that the set of dual frames determines a frame uniquely. Furthermore, for a given frame, the union of all coefficients of its dual frames is dense in ℓ2. We also introduce a class of frames (called pseudo-coherent frames), which includes Gabor frames and coherent frames, and investigate invertible pseudo-coherent frame multipliers, allowing a classification for frame-type operators for these frames. Finally, we give a numerical example for the invertibility of multipliers in the Gabor case.

We analyze the construction of a sequence space Θ, resp. a sequence of sequence spaces, in order to have {g i } ∞ i=1 as a Θ-frame or Banach frame for a Banach space X, resp. pre-F-frame or F-frame for a Fréchet space X F = ∩ s∈N0 X s , where {X s } s∈N0 is a sequence of Banach spaces.