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A Characterization of Schauder Frames Which Are Near-Schauder Bases

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Abstract

A basic problem of interest in connection with the study of Schauder frames in Banach spaces is that of characterizing those Schauder frames which can essentially be regarded as Schauder bases. In this paper, we give a solution to this problem using the notion of the minimal-associated sequence spaces and the minimal-associated reconstruction operators for Schauder frames. We prove that a Schauder frame is a near-Schauder basis if and only if the kernel of the minimal-associated reconstruction operator contains no copy of c 0. In particular, a Schauder frame of a Banach space with no copy of c 0 is a near-Schauder basis if and only if the minimal-associated sequence space contains no copy of c 0. In these cases, the minimal-associated reconstruction operator has a finite dimensional kernel and the dimension of the kernel is exactly the excess of the near-Schauder basis. Using these results, we make related applications on Besselian frames and near-Riesz bases.

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Authors and Affiliations

  1. Department of Mathematics and LPMC, Nankai University, Tianjin, 300071, Peoples Republic of China

    Rui Liu

  2. Department of Mathematics, Texas A & M University, College Station, TX, 77843-3368, USA

    Rui Liu

  3. Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX, 78712-0257, USA

    Bentuo Zheng

Authors
  1. Rui Liu
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  2. Bentuo Zheng
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Correspondence to Rui Liu.

Additional information

Communicated by Peter G. Casazza.

R. Liu was supported by funds from the Linear Analysis Workshop at Texas A&M University in 2008, the National Natural Science Foundation of China (No. 10571090), the Doctoral Programme Foundation of Institution of Higher Education (No. 20060055010), and the China Scholarship Council.

B. Zheng’s research is supported in part by NSF grant DMS-0800061.

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Liu, R., Zheng, B. A Characterization of Schauder Frames Which Are Near-Schauder Bases. J Fourier Anal Appl 16, 791–803 (2010). https://doi.org/10.1007/s00041-010-9126-5

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  • DOI: https://doi.org/10.1007/s00041-010-9126-5

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