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Homeomorphic images of orthogonal bases. (English) Zbl 1355.46017

Summary: Necessary and sufficient conditions are obtained for a sequence \(\{x_j: j \in \mathbb J\}\) in a Hilbert space to be, up to the elimination of a finite subset of \(\mathbb J\), the linear homeomorphic image of an orthogonal basis of some Hilbert space \(K\). This extends a similar result for orthonormal bases due to J. R. Holub [Proc. Am. Math. Soc. 122, No. 3, 779–785 (1994; Zbl 0821.46008)]. The proofs given here are based on simple linear algebra techniques.

MSC:

46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
42C15 General harmonic expansions, frames

Citations:

Zbl 0821.46008
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References:

[1] J.-P. Antonie and P. Balasz. Frames, semi-frames and Hilbert scales. Numer. Funct. Anal. Optimiz., 33:1–34, 2012.
[2] A. Askari-Hemmat, M.A. Dehghan, and M. Radjabalipour. Generalized frames and their redudancy. Proc. Amer. Math. Soc., 129(4):1143–1147, 2000. · Zbl 0976.42022
[3] H.H. Giv and M. Radjabalipour. On the structure and properties of lower bounded analytic frames. Iran. J. Sci. Technol. (IJST), 37(3):227–230, 2013.
[4] J.R. Holub. Pre-frame operators, Besselian frames, and near-Riesz bases in Hilbert spaces. Proc. Amer. Math. Soc., 122(3):779–785, 1994. Electronic Journal of Linear Algebra2016Homeomorphic Images of Orthogonal BasesM. KebryaeeM. RadjabalipourRecommended Citationtmp.1469409193.pdf.1v0He · Zbl 0821.46008
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