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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 243453, 2 pages
Comment on “Continuous g-Frame in Hilbert -Modules”
College of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
Received 1 April 2013; Accepted 16 April 2013
Copyright © 2013 Zhong-Qi Xiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The continuous g-frames in Hilbert -modules were introduced and investigated by Kouchi and Nazari (2011). They also studied the continuous g-Riesz basis and a characterization for it was presented by using the synthesis operator. However, we found that there is an error in the proof. The purpose of this paper is to improve their result by introducing the so-called modular continuous g-Riesz basis.
Kouchi and Nazari in  introduced the continuous g-frames in Hilbert -modules and investigated some of their properties. The following lemma is a useful tool in their study.
Lemma 1 (see ). Let be a -algebra, and two Hilbert -modules, and . The following statements are equivalent:(1) is surjective;(2) is bounded below with respect to norm, that is, there is such that for all ;(3) is bounded below with respect to inner product, that is, there is such that for all .
The authors also defined the continuous g-Riesz basis in Hilbert -modules as follows.
Definition 2. A continuous g-frame for Hilbert -module with respect to is said to be a continuous g-Riesz basis if it satisfies the following:(1) for any ;(2)if , then is equal to zero for each , where and is a measurable subset of .
By using the synthesis operator for a sequence defined by
they gave a characterization of continuous g-Riesz basis [1, Theorem 4.6].
Theorem 3. A family is a continuous g-Riesz basis for with respect to if and only if the synthesis operator is a homeomorphism.
We note, however, that in the proof of the above theorem, they said that “ for any , and , so ”, which is not true, because if has a dense range, then is one-to-one. We can improve their result by introducing the following modular continuous g-Riesz basis.
Definition 4. One calls a family in Hilbert -module a modular continuous g-Riesz basis if(1) ;(2)there exist constants such that for any ,
Theorem 5. A sequence is a modular continuous g-Riesz basis for with respect to if and only if the synthesis operator is a homeomorphism.
Proof. Suppose first that is a modular continuous g-Riesz basis for with synthesis operator . Then turns to be
showing that is bounded below with respect to norm. Hence, by Lemma 1, its adjoint operator is surjective. Since the condition (1) in Definition 4 implies that is injective, it follows that is invertible and so is invertible.
Conversely, let be a homeomorphism. Then is surjective, and again by Lemma 1, is injective. So the condition (1) in Definition 4 holds. Now for any ,
Therefore, is a modular continuous g-Riesz basis for with respect to .
- M. R. Kouchi and A. Nazari, “Continuous g-frame in Hilbert -modules,” Abstract and Applied Analysis, vol. 2011, Article ID 361595, 20 pages, 2011.
- L. Arambašić, “On frames for countably generated Hilbert -modules,” Proceedings of the American Mathematical Society, vol. 135, no. 2, pp. 469–478, 2007.