Let $ X $ and $ Y $ be two complex Banach spaces, $ {\mathbb C}^N $ the $ N $-dimensional complex Euclidean space with the inner product $ \langle z, w\rangle = \sum_{l = 1}^Nz_l\overline{w_l} $ and $ {\mathbb D}^N $ the unit polydisk in $ {\mathbb C}^N $. Let $ {\varphi} $ be a holomorphic self-map of $ {\mathbb D}^N $ and $ u\in H({\mathbb D}^N, \mathcal{L}(X, Y)) $, where $ H({\mathbb D}^N, X) $ denotes the space of all vector-valued holomorphic functions on $ {\mathbb D}^N $ and $ \mathcal{L}(X, Y) $ denotes the space of all bounded linear operators from $ X $ to $ Y $. The weighted composition operator $ W_{u, {\varphi}}:H({\mathbb D}^N, X)\to H({\mathbb D}^N, Y) $ is defined by
$ \begin{align*} W_{u,{\varphi}}f(z) = u(z)(f({\varphi}(z))). \end{align*} $
The bounded and compact weighted composition operators between vector-valued Bloch-type spaces on the unit polydisk are completely characterized in the paper.
Citation: Zhi-jie Jiang. Weighted composition operators between vector-valued Bloch-type spaces on the polydisk[J]. AIMS Mathematics, 2025, 10(8): 18957-18982. doi: 10.3934/math.2025847
Let $ X $ and $ Y $ be two complex Banach spaces, $ {\mathbb C}^N $ the $ N $-dimensional complex Euclidean space with the inner product $ \langle z, w\rangle = \sum_{l = 1}^Nz_l\overline{w_l} $ and $ {\mathbb D}^N $ the unit polydisk in $ {\mathbb C}^N $. Let $ {\varphi} $ be a holomorphic self-map of $ {\mathbb D}^N $ and $ u\in H({\mathbb D}^N, \mathcal{L}(X, Y)) $, where $ H({\mathbb D}^N, X) $ denotes the space of all vector-valued holomorphic functions on $ {\mathbb D}^N $ and $ \mathcal{L}(X, Y) $ denotes the space of all bounded linear operators from $ X $ to $ Y $. The weighted composition operator $ W_{u, {\varphi}}:H({\mathbb D}^N, X)\to H({\mathbb D}^N, Y) $ is defined by
$ \begin{align*} W_{u,{\varphi}}f(z) = u(z)(f({\varphi}(z))). \end{align*} $
The bounded and compact weighted composition operators between vector-valued Bloch-type spaces on the unit polydisk are completely characterized in the paper.
| [1] | R. F. Allen, F. Colonna, Weighted composition operators on the Bloch space of a bounded homogeneous domain, In: Topics in operator theory, Basel: Birkhäuser, 2010, 11–37. https://doi.org/10.1007/978-3-0346-0158-0_2 |
| [2] |
J. A. Cima, The basic properties of Bloch functions, Int. J. Math. Math. Sci., 2 (1979), 369–413. https://doi.org/10.1155/S0161171279000314 doi: 10.1155/S0161171279000314
|
| [3] | C. C. Cowen, B. D. Maccluer, Composition operators on spaces of analytic functions, New York: CRC Press, 1995. https://doi.org/10.1201/9781315139920 |
| [4] |
K. Esmaeili, H. Mahyar, Weighted composition operators between vector-valued Bloch-type spaces, Turk. J. Math., 43 (2019), 151–171. https://doi.org/10.3906/mat-1711-91 doi: 10.3906/mat-1711-91
|
| [5] |
Z.-S. Fang, Z.-H. Zhou, New characterizations of the weighted composition operators between Bloch type spaces in the polydisk, Can. Math. Bull., 57 (2014), 794–802. http://doi.org/10.4153/CMB-2013-043-4 doi: 10.4153/CMB-2013-043-4
|
| [6] |
F. Forelli, The isometries of $H^p$, Can. J. Math., 16 (1964), 721–728. https://doi.org/10.4153/CJM-1964-068-3 doi: 10.4153/CJM-1964-068-3
|
| [7] |
Z. J. Hu, Composition operators between Bloch-type spaces in polydisk, Science in China. Series A, Mathematics, Physics, Astronomy, 48 (2005), 268–282. https://doi.org/10.1360/ya2005-V48-IS1-P268-282 doi: 10.1360/ya2005-V48-IS1-P268-282
|
| [8] |
O. Hy${\rm{\ddot v}}$arinen, I. Nieminen, Weighted composition followed by differentiation between Bloch-type spaces, Rev. Mat. Complut., 27 (2014), 641–656. https://doi.org/10.1007/s13163-013-0138-y doi: 10.1007/s13163-013-0138-y
|
| [9] |
O. Hy${\rm{\ddot v}}$arinen, M. Lindström, Estimates of essential norms of weighted composition operators between Bloch-type spaces, J. Math. Anal. Appl., 393 (2012), 38–44. https://doi.org/10.1016/j.jmaa.2012.03.059 doi: 10.1016/j.jmaa.2012.03.059
|
| [10] |
S. G. Krantz, D. Ma, Bloch functions on strongly pseudoconvex domains, Indiana Univ. Math. J., 37 (1988), 145–163. https://doi.org/10.1512/iumj.1988.37.37007 doi: 10.1512/iumj.1988.37.37007
|
| [11] |
J. Laitila, Weakly compact composition operators on vector-valued BMOA, J. Math. Anal. Appl., 308 (2005), 730–745. https://doi.org/10.1016/j.jmaa.2004.12.002 doi: 10.1016/j.jmaa.2004.12.002
|
| [12] |
J. Laitila, H.-O. Tylli, Operator-weighted composition operators on vector-valued analytic function spaces, Illinois J. Math., 53 (2009), 1019–1032. https://doi.org/10.1215/ijm/1290435336 doi: 10.1215/ijm/1290435336
|
| [13] | J. Laitila, H.-O. Tylli, M. Wang, Composition operators from weak to strong spaces of vector-valued analytic functions. J. Operat. Theor., 62 (2009), 281–295. |
| [14] |
J. S. Manhas, R. Zhao, New estimates of essential norms of weighted composition operators between Bloch type spaces, J. Math. Anal. Appl., 389 (2012), 32–47. https://doi.org/10.1016/j.jmaa.2011.11.039 doi: 10.1016/j.jmaa.2011.11.039
|
| [15] |
B. D. Maccluer, R. Zhao, Essential norm of weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math., 33 (2003), 1437–1458. https://doi.org/10.1216/rmjm/1181075473 doi: 10.1216/rmjm/1181075473
|
| [16] | W. Rudin, Function theory in the unit ball of $ {\mathbb C}^n$, Berlin: Springer, 2008. https://doi.org/10.1007/978-3-540-68276-9 |
| [17] |
S. Stević, R. Chen, Z. Zhou, Weighted composition operators between Bloch-type spaces in the polydisk, Sb. Math., 201 (2010), 289–319. https://doi.org/10.1070/SM2010v201n02ABEH004073 doi: 10.1070/SM2010v201n02ABEH004073
|
| [18] | H. Xu, T. Liu, Weighted composition operators between different Bloch-type spaces on the unit polydisk, (Chinese), Chinese Annals of Mathematics Series B, 26A (2005), 61–72. |
| [19] | K. H. Zhu, Spaces of holomorphic functions in the unit ball, New York: Springer, 2005. https://doi.org/10.1007/0-387-27539-8 |
Zhi-jie Jiang. Weighted composition operators between vector-valued Bloch-type spaces on the polydisk[J]. AIMS Mathematics, 2025, 10(8): 18957-18982. doi: 10.3934/math.2025847
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