Syndetic Sensitivity and Mean Sensitivity for Linear Operators
We study syndetic sensitivity and mean sensitivity for linear dynamical systems. For the syndetic sensitivity aspect, we obtain some properties of syndetic sensitivity for adjoint operators and left multiplication operators. We also show that there exists a linear dynamical system <inline-formula...
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| Format: | Article |
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MDPI AG
2023-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/13/2796 |
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| author | Quanquan Yao Peiyong Zhu |
| author_facet | Quanquan Yao Peiyong Zhu |
| author_sort | Quanquan Yao |
| collection | DOAJ |
| description | We study syndetic sensitivity and mean sensitivity for linear dynamical systems. For the syndetic sensitivity aspect, we obtain some properties of syndetic sensitivity for adjoint operators and left multiplication operators. We also show that there exists a linear dynamical system <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo>,</mo><mi>T</mi><mo>×</mo><mi>S</mi><mo>)</mo></mrow></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo>,</mo><mi>T</mi><mo>×</mo><mi>S</mi><mo>)</mo></mrow></semantics></math></inline-formula> is cofinitely sensitive but <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>Y</mi><mo>,</mo><mi>S</mi><mo>)</mo></mrow></semantics></math></inline-formula> are not syndetically sensitive. For the mean sensitivity aspect, we show that if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>Y</mi><mo>,</mo><mi>S</mi><mo>)</mo></mrow></semantics></math></inline-formula> is sensitive and not mean sensitive, where <i>Y</i> is a complex Banach space, the spectrum of <i>T</i> meets the unit circle. We also obtain some results regarding mean sensitive perturbations. |
| first_indexed | 2024-03-11T01:35:19Z |
| format | Article |
| id | doaj.art-8219cb2c24ef4543953c631009d980c4 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T01:35:19Z |
| publishDate | 2023-06-01 |
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| series | Mathematics |
| spelling | doaj.art-8219cb2c24ef4543953c631009d980c42023-11-18T17:01:27ZengMDPI AGMathematics2227-73902023-06-011113279610.3390/math11132796Syndetic Sensitivity and Mean Sensitivity for Linear OperatorsQuanquan Yao0Peiyong Zhu1School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, ChinaSchool of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, ChinaWe study syndetic sensitivity and mean sensitivity for linear dynamical systems. For the syndetic sensitivity aspect, we obtain some properties of syndetic sensitivity for adjoint operators and left multiplication operators. We also show that there exists a linear dynamical system <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo>,</mo><mi>T</mi><mo>×</mo><mi>S</mi><mo>)</mo></mrow></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo>,</mo><mi>T</mi><mo>×</mo><mi>S</mi><mo>)</mo></mrow></semantics></math></inline-formula> is cofinitely sensitive but <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>Y</mi><mo>,</mo><mi>S</mi><mo>)</mo></mrow></semantics></math></inline-formula> are not syndetically sensitive. For the mean sensitivity aspect, we show that if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>Y</mi><mo>,</mo><mi>S</mi><mo>)</mo></mrow></semantics></math></inline-formula> is sensitive and not mean sensitive, where <i>Y</i> is a complex Banach space, the spectrum of <i>T</i> meets the unit circle. We also obtain some results regarding mean sensitive perturbations.https://www.mdpi.com/2227-7390/11/13/2796syndetically sensitivecofinitely sensitivemean sensitive |
| spellingShingle | Quanquan Yao Peiyong Zhu Syndetic Sensitivity and Mean Sensitivity for Linear Operators Mathematics syndetically sensitive cofinitely sensitive mean sensitive |
| title | Syndetic Sensitivity and Mean Sensitivity for Linear Operators |
| title_full | Syndetic Sensitivity and Mean Sensitivity for Linear Operators |
| title_fullStr | Syndetic Sensitivity and Mean Sensitivity for Linear Operators |
| title_full_unstemmed | Syndetic Sensitivity and Mean Sensitivity for Linear Operators |
| title_short | Syndetic Sensitivity and Mean Sensitivity for Linear Operators |
| title_sort | syndetic sensitivity and mean sensitivity for linear operators |
| topic | syndetically sensitive cofinitely sensitive mean sensitive |
| url | https://www.mdpi.com/2227-7390/11/13/2796 |
| work_keys_str_mv | AT quanquanyao syndeticsensitivityandmeansensitivityforlinearoperators AT peiyongzhu syndeticsensitivityandmeansensitivityforlinearoperators |