Hilbert space frames containing a Riesz basis and Banach spaces which have no subspace isomorphic to c0c_0

We prove that a Hilbert space frame \fti contains a Riesz basis if every subfamily \ftj , J \subseteq I , is a frame for its closed span. Secondly we give a new characterization of Banach spaces which do not have any subspace isomorphic to $c_0$. This result immediately leads to an improvement of a recent theorem of Holub concerning frames consisting of a Riesz basis plus finitely many elements

On R-duals and the duality principle in Gabor analysis

The concept of R-duals of a frame was introduced by Casazza, Kutyniok and Lammers in 2004, with the motivation to obtain a general version of the duality principle in Gabor analysis. For tight Gabor frames and Gabor Riesz bases the three authors were actually able to show that the duality principle is a special case of general results for R-duals. In this paper we introduce various alternative R-duals, with focus on what we call R-duals of type II and III. We show how they are related and provide characterizations of the R-duals of type II and III. In particular, we prove that for tight frames these classes coincide with the R-duals by Casazza et el., which is desirable in the sense that the motivating case of tight Gabor frames already is well covered by these R-duals. On the other hand, all the introduced types of R-duals generalize the duality principle for larger classes of Gabor frames than just the tight frames and the Riesz bases; in particular, the R-duals of type III cover the duality principle for all Gabor frames

Frames for the solution of operator equations in Hilbert spaces with fixed dual pairing

For the solution of operator equations, Stevenson introduced a definition of frames, where a Hilbert space and its dual are {\em not} identified. This means that the Riesz isomorphism is not used as an identification, which, for example, does not make sense for the Sobolev spaces $H_0^1(\Omega)$ and $H^{-1}(\Omega)$. In this article, we are going to revisit the concept of Stevenson frames and introduce it for Banach spaces. This is equivalent to $\ell^2$-Banach frames. It is known that, if such a system exists, by defining a new inner product and using the Riesz isomorphism, the Banach space is isomorphic to a Hilbert space. In this article, we deal with the contrasting setting, where $\mathcal H$ and $\mathcal H'$ are not identified, and equivalent norms are distinguished, and show that in this setting the investigation of $\ell^2$-Banach frames make sense.Comment: 23 pages; accepted for publication in 'Numerical Functional Analysis and Optimization

Operator representations of frames: boundedness, duality, and stability

The purpose of the paper is to analyze frames $\{f_k\}_{k\in \mathbf Z}$ having the form $\{T^kf_0\}_{k\in\mathbf Z}$ for some linear operator T: \mbox{span} \{f_k\}_{k\in \mathbf Z} \to \mbox{span}\{f_k\}_{k\in \mathbf Z}. A key result characterizes boundedness of the operator $T$ in terms of shift-invariance of a certain sequence space. One of the consequences is a characterization of the case where the representation $\{f_k\}_{k\in \mathbf Z}=\{T^kf_0\}_{k\in\mathbf Z}$ can be achieved for an operator $T$ that has an extension to a bounded bijective operator $\widetilde{T}: \cal H \to \cal H.$ In this case we also characterize all the dual frames that are representable in terms of iterations of an operator $V;$ in particular we prove that the only possible operator is $V=(\widetilde{T}^*)^{-1}.$ Finally, we consider stability of the representation $\{T^kf_0\}_{k\in\mathbf Z};$ rather surprisingly, it turns out that the possibility to represent a frame on this form is sensitive towards some of the classical perturbation conditions in frame theory. Various ways of avoiding this problem will be discussed. Throughout the paper the results will be connected with the operators and function systems appearing in applied harmonic analysis, as well as with general group representations