$K$-$b$-Frames for Hilbert Spaces and the $b$-Adjoint Operator
This paper will generalize $b$-frames, a new concept of frames for Hilbert spaces, by $K$-$b$-frames. The idea is to take a sequence from a Banach space and see how it can be a frame for a Hilbert space. Instead of the scalar product, we will use a new product called the $b$-dual product, which is c...
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| Language: | English |
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University of Maragheh
2024-10-01
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| Series: | Sahand Communications in Mathematical Analysis |
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| Online Access: | https://scma.maragheh.ac.ir/article_713024_0d90f228ed533eeb3746d7183c505bfe.pdf |
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| author | Chaimae Mezzat Samir Kabbaj |
| author_facet | Chaimae Mezzat Samir Kabbaj |
| author_sort | Chaimae Mezzat |
| collection | DOAJ |
| description | This paper will generalize $b$-frames, a new concept of frames for Hilbert spaces, by $K$-$b$-frames. The idea is to take a sequence from a Banach space and see how it can be a frame for a Hilbert space. Instead of the scalar product, we will use a new product called the $b$-dual product, which is constructed via a bilinear mapping. We will introduce new results about this product, about $b$-frames and $K$-$b$-frames and we will also give some examples of both $b$-frames and $K$-$b$-frames that have never been given before. We will express the reconstruction formula of the elements of the Hilbert space. We will also study the stability and preservation of both $b$-frames and $K$-$b$-frames and to do so, we will give the equivalent of the adjoint operator according to the $b$-dual product. |
| format | Article |
| id | doaj-art-735b8d07f24e4adb867ce31a1a113ae3 |
| institution | Kabale University |
| issn | 2322-5807 2423-3900 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | University of Maragheh |
| record_format | Article |
| series | Sahand Communications in Mathematical Analysis |
| spelling | doaj-art-735b8d07f24e4adb867ce31a1a113ae32025-02-11T05:27:48ZengUniversity of MaraghehSahand Communications in Mathematical Analysis2322-58072423-39002024-10-0121412610.22130/scma.2023.2012970.1485713024$K$-$b$-Frames for Hilbert Spaces and the $b$-Adjoint OperatorChaimae Mezzat0Samir Kabbaj1Department of Mathematics, Faculty of Science, Ibn Tofail University, B.P. 133, Kenitra, Morocco.Department of Mathematics, Faculty of Science, Ibn Tofail University, B.P. 133, Kenitra, Morocco.This paper will generalize $b$-frames, a new concept of frames for Hilbert spaces, by $K$-$b$-frames. The idea is to take a sequence from a Banach space and see how it can be a frame for a Hilbert space. Instead of the scalar product, we will use a new product called the $b$-dual product, which is constructed via a bilinear mapping. We will introduce new results about this product, about $b$-frames and $K$-$b$-frames and we will also give some examples of both $b$-frames and $K$-$b$-frames that have never been given before. We will express the reconstruction formula of the elements of the Hilbert space. We will also study the stability and preservation of both $b$-frames and $K$-$b$-frames and to do so, we will give the equivalent of the adjoint operator according to the $b$-dual product.https://scma.maragheh.ac.ir/article_713024_0d90f228ed533eeb3746d7183c505bfe.pdfb-framesk-b-framesb-adjointhilbert spacesesquilinear form |
| spellingShingle | Chaimae Mezzat Samir Kabbaj $K$-$b$-Frames for Hilbert Spaces and the $b$-Adjoint Operator Sahand Communications in Mathematical Analysis b-frames k-b-frames b-adjoint hilbert space sesquilinear form |
| title | $K$-$b$-Frames for Hilbert Spaces and the $b$-Adjoint Operator |
| title_full | $K$-$b$-Frames for Hilbert Spaces and the $b$-Adjoint Operator |
| title_fullStr | $K$-$b$-Frames for Hilbert Spaces and the $b$-Adjoint Operator |
| title_full_unstemmed | $K$-$b$-Frames for Hilbert Spaces and the $b$-Adjoint Operator |
| title_short | $K$-$b$-Frames for Hilbert Spaces and the $b$-Adjoint Operator |
| title_sort | k b frames for hilbert spaces and the b adjoint operator |
| topic | b-frames k-b-frames b-adjoint hilbert space sesquilinear form |
| url | https://scma.maragheh.ac.ir/article_713024_0d90f228ed533eeb3746d7183c505bfe.pdf |
| work_keys_str_mv | AT chaimaemezzat kbframesforhilbertspacesandthebadjointoperator AT samirkabbaj kbframesforhilbertspacesandthebadjointoperator |