A matrix acting between Fock spaces
Abstract If H ν = ( ν n , k ) n , k ≥ 0 $\mathcal{H}_{\nu}=(\nu _{n,k})_{n,k\geq 0}$ is the matrix with entries ν n , k = ∫ [ 0 , ∞ ) t n + k n ! d ν ( t ) $\nu _{n,k}=\int _{[0,\infty )}\frac{ t^{n+k}}{n!}\,d\nu (t)$ , where ν is a nonnegative Borel measure on the interval [ 0 , ∞ ) $[0,\infty )$ ,...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2024-01-01
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| Series: | Journal of Inequalities and Applications |
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| Online Access: | https://doi.org/10.1186/s13660-024-03084-7 |
| _version_ | 1797276210591432704 |
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| author | Zhengyuan Zhuo Dongxing Li Tiaoying Zeng |
| author_facet | Zhengyuan Zhuo Dongxing Li Tiaoying Zeng |
| author_sort | Zhengyuan Zhuo |
| collection | DOAJ |
| description | Abstract If H ν = ( ν n , k ) n , k ≥ 0 $\mathcal{H}_{\nu}=(\nu _{n,k})_{n,k\geq 0}$ is the matrix with entries ν n , k = ∫ [ 0 , ∞ ) t n + k n ! d ν ( t ) $\nu _{n,k}=\int _{[0,\infty )}\frac{ t^{n+k}}{n!}\,d\nu (t)$ , where ν is a nonnegative Borel measure on the interval [ 0 , ∞ ) $[0,\infty )$ , the matrix H ν $\mathcal{H}_{\nu}$ acts on the space of all entire functions f ( z ) = ∑ n = 0 ∞ a n z n $f(z) =\sum_{n=0}^{\infty} a_{n} z^{n}$ and induces formally the operator in the following way: H ν ( f ) ( z ) = ∑ n = 0 ∞ ( ∑ k = 0 ∞ ν n , k a k ) z n . $$ \mathcal{H}_{\nu}(f) (z)=\sum_{n=0}^{\infty} \Biggl(\sum_{k=0}^{\infty}\nu _{n,k}a_{k}\Biggr)z^{n}. $$ In this paper, for 0 < p ≤ ∞ $0< p\leq \infty $ , we classify for which measures the operator H ν ( f ) $\mathcal{H}_{\nu}(f)$ is well defined on F p $F^{p}$ and also gets an integral representation, and among them we characterize those for which H ν $\mathcal{H}_{\nu}$ is a bounded (resp., compact) operator between F p $F^{p} $ and F ∞ $F^{\infty }$ . |
| first_indexed | 2024-03-07T15:25:06Z |
| format | Article |
| id | doaj.art-b298dd3ed062487a835549209739caec |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-03-07T15:25:06Z |
| publishDate | 2024-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-b298dd3ed062487a835549209739caec2024-03-05T17:08:56ZengSpringerOpenJournal of Inequalities and Applications1029-242X2024-01-012024111610.1186/s13660-024-03084-7A matrix acting between Fock spacesZhengyuan Zhuo0Dongxing Li1Tiaoying Zeng2School of Mathematics and Systems Science, Guangdong Polytechnic Normal UniversitySchool of Financial Mathematics and Statistics, Guangdong University of FinanceSchool of Mathematics, Jiaying UniversityAbstract If H ν = ( ν n , k ) n , k ≥ 0 $\mathcal{H}_{\nu}=(\nu _{n,k})_{n,k\geq 0}$ is the matrix with entries ν n , k = ∫ [ 0 , ∞ ) t n + k n ! d ν ( t ) $\nu _{n,k}=\int _{[0,\infty )}\frac{ t^{n+k}}{n!}\,d\nu (t)$ , where ν is a nonnegative Borel measure on the interval [ 0 , ∞ ) $[0,\infty )$ , the matrix H ν $\mathcal{H}_{\nu}$ acts on the space of all entire functions f ( z ) = ∑ n = 0 ∞ a n z n $f(z) =\sum_{n=0}^{\infty} a_{n} z^{n}$ and induces formally the operator in the following way: H ν ( f ) ( z ) = ∑ n = 0 ∞ ( ∑ k = 0 ∞ ν n , k a k ) z n . $$ \mathcal{H}_{\nu}(f) (z)=\sum_{n=0}^{\infty} \Biggl(\sum_{k=0}^{\infty}\nu _{n,k}a_{k}\Biggr)z^{n}. $$ In this paper, for 0 < p ≤ ∞ $0< p\leq \infty $ , we classify for which measures the operator H ν ( f ) $\mathcal{H}_{\nu}(f)$ is well defined on F p $F^{p}$ and also gets an integral representation, and among them we characterize those for which H ν $\mathcal{H}_{\nu}$ is a bounded (resp., compact) operator between F p $F^{p} $ and F ∞ $F^{\infty }$ .https://doi.org/10.1186/s13660-024-03084-7Fock spacesMatricesFock Carleson measure |
| spellingShingle | Zhengyuan Zhuo Dongxing Li Tiaoying Zeng A matrix acting between Fock spaces Journal of Inequalities and Applications Fock spaces Matrices Fock Carleson measure |
| title | A matrix acting between Fock spaces |
| title_full | A matrix acting between Fock spaces |
| title_fullStr | A matrix acting between Fock spaces |
| title_full_unstemmed | A matrix acting between Fock spaces |
| title_short | A matrix acting between Fock spaces |
| title_sort | matrix acting between fock spaces |
| topic | Fock spaces Matrices Fock Carleson measure |
| url | https://doi.org/10.1186/s13660-024-03084-7 |
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