The Backward Shift and Two Infinite-Dimension Phenomena in Banach Spaces
We consider the backward shift operator on a sequence Banach space in the context of two infinite-dimensional phenomena: the existence of topologically transitive operators, and the existence of entire analytic functions of the unbounded type. It is well known that the weighted backward shift (for a...
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| Format: | Article |
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MDPI AG
2023-10-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/10/1855 |
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| author | Zoriana Novosad Andriy Zagorodnyuk |
| author_facet | Zoriana Novosad Andriy Zagorodnyuk |
| author_sort | Zoriana Novosad |
| collection | DOAJ |
| description | We consider the backward shift operator on a sequence Banach space in the context of two infinite-dimensional phenomena: the existence of topologically transitive operators, and the existence of entire analytic functions of the unbounded type. It is well known that the weighted backward shift (for an appropriated weight) is topologically transitive on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>p</mi><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula> and on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>c</mi><mn>0</mn></msub><mo>.</mo></mrow></semantics></math></inline-formula> We construct some generalizations of the weighted backward shift for non-separable Banach spaces, which remains topologically transitive. Also, we show that the backward shift, in some sense, generates analytic functions of the unbounded type. We introduce the notion of a generator of analytic functions of the unbounded type on a Banach space and investigate its properties. In addition, we show that, using this operator, one can obtain a quasi-extension operator of analytic functions in a germ of zero for entire analytic functions. The results are supported by examples. |
| first_indexed | 2024-03-10T20:52:20Z |
| format | Article |
| id | doaj.art-fb81558b7c6b4c04bff6cf8d0a78cdb1 |
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| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T20:52:20Z |
| publishDate | 2023-10-01 |
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| series | Symmetry |
| spelling | doaj.art-fb81558b7c6b4c04bff6cf8d0a78cdb12023-11-19T18:17:45ZengMDPI AGSymmetry2073-89942023-10-011510185510.3390/sym15101855The Backward Shift and Two Infinite-Dimension Phenomena in Banach SpacesZoriana Novosad0Andriy Zagorodnyuk1Department of Higher Mathematics and Quantitative Methods 10, Lviv University of Trade and Economics, Tuhan-Baranovsky Str., 79005 Lviv, UkraineFaculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University, 57 Shevchenka Str., 76018 Ivano-Frankivsk, UkraineWe consider the backward shift operator on a sequence Banach space in the context of two infinite-dimensional phenomena: the existence of topologically transitive operators, and the existence of entire analytic functions of the unbounded type. It is well known that the weighted backward shift (for an appropriated weight) is topologically transitive on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>p</mi><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula> and on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>c</mi><mn>0</mn></msub><mo>.</mo></mrow></semantics></math></inline-formula> We construct some generalizations of the weighted backward shift for non-separable Banach spaces, which remains topologically transitive. Also, we show that the backward shift, in some sense, generates analytic functions of the unbounded type. We introduce the notion of a generator of analytic functions of the unbounded type on a Banach space and investigate its properties. In addition, we show that, using this operator, one can obtain a quasi-extension operator of analytic functions in a germ of zero for entire analytic functions. The results are supported by examples.https://www.mdpi.com/2073-8994/15/10/1855topologically transitive operatorsanalytic functions on Banach spacesanalytic functions of unbounded type |
| spellingShingle | Zoriana Novosad Andriy Zagorodnyuk The Backward Shift and Two Infinite-Dimension Phenomena in Banach Spaces Symmetry topologically transitive operators analytic functions on Banach spaces analytic functions of unbounded type |
| title | The Backward Shift and Two Infinite-Dimension Phenomena in Banach Spaces |
| title_full | The Backward Shift and Two Infinite-Dimension Phenomena in Banach Spaces |
| title_fullStr | The Backward Shift and Two Infinite-Dimension Phenomena in Banach Spaces |
| title_full_unstemmed | The Backward Shift and Two Infinite-Dimension Phenomena in Banach Spaces |
| title_short | The Backward Shift and Two Infinite-Dimension Phenomena in Banach Spaces |
| title_sort | backward shift and two infinite dimension phenomena in banach spaces |
| topic | topologically transitive operators analytic functions on Banach spaces analytic functions of unbounded type |
| url | https://www.mdpi.com/2073-8994/15/10/1855 |
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