2-Complex Symmetric Composition Operators on <i>H</i><sup>2</sup>
In this paper, we study 2-complex symmetric composition operators with the conjugation <i>J</i>, defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mi>f<...
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MDPI AG
2022-07-01
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| Online Access: | https://www.mdpi.com/2075-1680/11/8/358 |
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| author | Lian Hu Songxiao Li Rong Yang |
| author_facet | Lian Hu Songxiao Li Rong Yang |
| author_sort | Lian Hu |
| collection | DOAJ |
| description | In this paper, we study 2-complex symmetric composition operators with the conjugation <i>J</i>, defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mover><mrow><mo>(</mo><mi>f</mi><mrow><mo>(</mo><mover accent="true"><mi>z</mi><mo stretchy="false">¯</mo></mover><mo>)</mo></mrow><mo>)</mo></mrow><mo stretchy="false">¯</mo></mover></mrow></semantics></math></inline-formula>, on the Hardy space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mn>2</mn></msup></semantics></math></inline-formula>. More precisely, we obtain the necessary and sufficient condition for the composition operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>C</mi><mi>ϕ</mi></msub></semantics></math></inline-formula> to be 2-complex symmetric with <i>J</i> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula> is an automorphism of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">D</mi></semantics></math></inline-formula>. We also characterize 2-complex symmetric with <i>J</i> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula> is a linear fractional self-map of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">D</mi></semantics></math></inline-formula>. |
| first_indexed | 2024-03-09T04:41:54Z |
| format | Article |
| id | doaj.art-65adc41569174368b14dec9bef8d3b64 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-09T04:41:54Z |
| publishDate | 2022-07-01 |
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| spelling | doaj.art-65adc41569174368b14dec9bef8d3b642023-12-03T13:19:53ZengMDPI AGAxioms2075-16802022-07-0111835810.3390/axioms110803582-Complex Symmetric Composition Operators on <i>H</i><sup>2</sup>Lian Hu0Songxiao Li1Rong Yang2Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610054, ChinaInstitute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610054, ChinaInstitute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610054, ChinaIn this paper, we study 2-complex symmetric composition operators with the conjugation <i>J</i>, defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mover><mrow><mo>(</mo><mi>f</mi><mrow><mo>(</mo><mover accent="true"><mi>z</mi><mo stretchy="false">¯</mo></mover><mo>)</mo></mrow><mo>)</mo></mrow><mo stretchy="false">¯</mo></mover></mrow></semantics></math></inline-formula>, on the Hardy space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mn>2</mn></msup></semantics></math></inline-formula>. More precisely, we obtain the necessary and sufficient condition for the composition operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>C</mi><mi>ϕ</mi></msub></semantics></math></inline-formula> to be 2-complex symmetric with <i>J</i> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula> is an automorphism of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">D</mi></semantics></math></inline-formula>. We also characterize 2-complex symmetric with <i>J</i> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula> is a linear fractional self-map of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">D</mi></semantics></math></inline-formula>.https://www.mdpi.com/2075-1680/11/8/358composition operatorm-complex symmetricnormal |
| spellingShingle | Lian Hu Songxiao Li Rong Yang 2-Complex Symmetric Composition Operators on <i>H</i><sup>2</sup> Axioms composition operator m-complex symmetric normal |
| title | 2-Complex Symmetric Composition Operators on <i>H</i><sup>2</sup> |
| title_full | 2-Complex Symmetric Composition Operators on <i>H</i><sup>2</sup> |
| title_fullStr | 2-Complex Symmetric Composition Operators on <i>H</i><sup>2</sup> |
| title_full_unstemmed | 2-Complex Symmetric Composition Operators on <i>H</i><sup>2</sup> |
| title_short | 2-Complex Symmetric Composition Operators on <i>H</i><sup>2</sup> |
| title_sort | 2 complex symmetric composition operators on i h i sup 2 sup |
| topic | composition operator m-complex symmetric normal |
| url | https://www.mdpi.com/2075-1680/11/8/358 |
| work_keys_str_mv | AT lianhu 2complexsymmetriccompositionoperatorsonihisup2sup AT songxiaoli 2complexsymmetriccompositionoperatorsonihisup2sup AT rongyang 2complexsymmetriccompositionoperatorsonihisup2sup |