Complex symmetric Toeplitz operators on the generalized derivative Hardy space
Abstract The generalized derivative Hardy space S α , β 2 ( D ) $S^{2}_{\alpha ,\beta}(\mathbb{D})$ consists of all functions whose derivatives are in the Hardy and Bergman spaces as follows: for positive integers α, β, S α , β 2 ( D ) = { f ∈ H ( D ) : ∥ f ∥ S α , β 2 2 = ∥ f ∥ H 2 2 + α + β α β ∥...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2022-06-01
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| Series: | Journal of Inequalities and Applications |
| Online Access: | https://doi.org/10.1186/s13660-022-02810-3 |
| _version_ | 1818553412367679488 |
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| author | Eungil Ko Ji Eun Lee Jongrak Lee |
| author_facet | Eungil Ko Ji Eun Lee Jongrak Lee |
| author_sort | Eungil Ko |
| collection | DOAJ |
| description | Abstract The generalized derivative Hardy space S α , β 2 ( D ) $S^{2}_{\alpha ,\beta}(\mathbb{D})$ consists of all functions whose derivatives are in the Hardy and Bergman spaces as follows: for positive integers α, β, S α , β 2 ( D ) = { f ∈ H ( D ) : ∥ f ∥ S α , β 2 2 = ∥ f ∥ H 2 2 + α + β α β ∥ f ′ ∥ A 2 2 + 1 α β ∥ f ′ ∥ H 2 2 < ∞ } , $$ S^{2}_{\alpha ,\beta}(\mathbb{D})= \biggl\{ f\in H(\mathbb{D}) : \Vert {f} \Vert ^{2}_{S^{2}_{ \alpha ,\beta}}= \Vert {f} \Vert ^{2}_{H^{2}}+{ \frac{{\alpha +\beta}}{\alpha \beta}} \bigl\Vert {f'} \bigr\Vert ^{2}_{A^{2}}+ \frac{1}{\alpha \beta} \bigl\Vert {f'} \bigr\Vert ^{2}_{H^{2}}< \infty \biggr\} , $$ where H ( D ) $H({\mathbb{D}})$ denotes the space of all functions analytic on the open unit disk D ${\mathbb{D}}$ . In this paper, we study characterizations for Toeplitz operators to be complex symmetric on the generalized derivative Hardy space S α , β 2 ( D ) $S^{2}_{\alpha ,\beta}(\mathbb{D})$ with respect to some conjugations C ξ $C_{\xi}$ , C μ , λ $C_{\mu , \lambda}$ . Moreover, for any conjugation C, we consider the necessary and sufficient conditions for complex symmetric Toeplitz operators with the symbol φ of the form φ ( z ) = ∑ n = 1 ∞ φ ˆ ( − n ) ‾ z ‾ n + ∑ n = 0 ∞ φ ˆ ( n ) z n $\varphi (z)=\sum_{n=1}^{\infty}\overline{\hat{\varphi}(-n)} \overline{z}^{n}+\sum_{n=0}^{\infty}\hat{\varphi}(n)z^{n}$ . Next, we also study complex symmetric Toeplitz operators with non-harmonic symbols on the generalized derivative Hardy space S α , β 2 ( D ) $S^{2}_{\alpha ,\beta}(\mathbb{D})$ . |
| first_indexed | 2024-12-12T09:25:30Z |
| format | Article |
| id | doaj.art-1718ef0a57ec4cd386a898d1d0935340 |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-12T09:25:30Z |
| publishDate | 2022-06-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-1718ef0a57ec4cd386a898d1d09353402022-12-22T00:29:03ZengSpringerOpenJournal of Inequalities and Applications1029-242X2022-06-012022111210.1186/s13660-022-02810-3Complex symmetric Toeplitz operators on the generalized derivative Hardy spaceEungil Ko0Ji Eun Lee1Jongrak Lee2Department of Mathematics, Ewha Womans UniversityDepartment of Mathematics and Statistics, Sejong UniversityDepartment of Mathematics, Sungkyunkwan UniversityAbstract The generalized derivative Hardy space S α , β 2 ( D ) $S^{2}_{\alpha ,\beta}(\mathbb{D})$ consists of all functions whose derivatives are in the Hardy and Bergman spaces as follows: for positive integers α, β, S α , β 2 ( D ) = { f ∈ H ( D ) : ∥ f ∥ S α , β 2 2 = ∥ f ∥ H 2 2 + α + β α β ∥ f ′ ∥ A 2 2 + 1 α β ∥ f ′ ∥ H 2 2 < ∞ } , $$ S^{2}_{\alpha ,\beta}(\mathbb{D})= \biggl\{ f\in H(\mathbb{D}) : \Vert {f} \Vert ^{2}_{S^{2}_{ \alpha ,\beta}}= \Vert {f} \Vert ^{2}_{H^{2}}+{ \frac{{\alpha +\beta}}{\alpha \beta}} \bigl\Vert {f'} \bigr\Vert ^{2}_{A^{2}}+ \frac{1}{\alpha \beta} \bigl\Vert {f'} \bigr\Vert ^{2}_{H^{2}}< \infty \biggr\} , $$ where H ( D ) $H({\mathbb{D}})$ denotes the space of all functions analytic on the open unit disk D ${\mathbb{D}}$ . In this paper, we study characterizations for Toeplitz operators to be complex symmetric on the generalized derivative Hardy space S α , β 2 ( D ) $S^{2}_{\alpha ,\beta}(\mathbb{D})$ with respect to some conjugations C ξ $C_{\xi}$ , C μ , λ $C_{\mu , \lambda}$ . Moreover, for any conjugation C, we consider the necessary and sufficient conditions for complex symmetric Toeplitz operators with the symbol φ of the form φ ( z ) = ∑ n = 1 ∞ φ ˆ ( − n ) ‾ z ‾ n + ∑ n = 0 ∞ φ ˆ ( n ) z n $\varphi (z)=\sum_{n=1}^{\infty}\overline{\hat{\varphi}(-n)} \overline{z}^{n}+\sum_{n=0}^{\infty}\hat{\varphi}(n)z^{n}$ . Next, we also study complex symmetric Toeplitz operators with non-harmonic symbols on the generalized derivative Hardy space S α , β 2 ( D ) $S^{2}_{\alpha ,\beta}(\mathbb{D})$ .https://doi.org/10.1186/s13660-022-02810-3 |
| spellingShingle | Eungil Ko Ji Eun Lee Jongrak Lee Complex symmetric Toeplitz operators on the generalized derivative Hardy space Journal of Inequalities and Applications |
| title | Complex symmetric Toeplitz operators on the generalized derivative Hardy space |
| title_full | Complex symmetric Toeplitz operators on the generalized derivative Hardy space |
| title_fullStr | Complex symmetric Toeplitz operators on the generalized derivative Hardy space |
| title_full_unstemmed | Complex symmetric Toeplitz operators on the generalized derivative Hardy space |
| title_short | Complex symmetric Toeplitz operators on the generalized derivative Hardy space |
| title_sort | complex symmetric toeplitz operators on the generalized derivative hardy space |
| url | https://doi.org/10.1186/s13660-022-02810-3 |
| work_keys_str_mv | AT eungilko complexsymmetrictoeplitzoperatorsonthegeneralizedderivativehardyspace AT jieunlee complexsymmetrictoeplitzoperatorsonthegeneralizedderivativehardyspace AT jongraklee complexsymmetrictoeplitzoperatorsonthegeneralizedderivativehardyspace |