×

Composition operators on the Bergman spaces of a minimal bounded homogeneous domain. (English) Zbl 1304.47034

Let \(\mu\) be a minimal bounded homogeneous domain in \(\mathbb C^d\), \(dV\) the Lebesgue measure on \(\mathbb C^d\) and \( o( \mu )\) the space of all holomorphic functions on \(\mu\). The Bergman kernel \(K_\mu :\mu \times \mu \to\mathbb C\) is the reproducing kernel of the Bergman space \( L_a^2 ( {\mu ,dV} ): = L^2 ( {\mu ,dV} ) \cap o( \mu )\). For \( \beta \in\mathbb R\), let \(dV_\beta\) denote the measure on \(\mu\) given by \( dV_\beta ( z ): = K_\mu ( {z,z} )^{ - \beta } dV( z )\). It is known that there exists a constant \( \varepsilon _{\min }\) such that \( L_a^p ( {\mu ,dV_\beta } )\) is nontrivial for all \(p\) if \(\beta > \varepsilon _{\min }\).
Every holomorphic map \( \varphi :\mu \to \mu\) defines a composition operator \(C_\varphi\) on \(o( \mu )\), in particular on \( L_a^p ( {\mu ,dV_\beta } )\), by \(C_\varphi f: = f \circ \varphi\). Thanks to properties of Carleson measures, the author obtains the following result:
If \(C_\varphi\) is a bounded (resp., compact) operator on \(L_a^q ( {\mu ,dV_{\beta _0 } } )\) for some \( q > 0\) and \( \beta _0 > \varepsilon _{\min }\), then \(C_\varphi\) is a bounded (resp., compact) operator on \(L_a^p ( {\mu ,dV_{\beta } } )\) for any \( p > 0\) and \( \beta \geq {\beta _0 }\).
Composition operators on the weighted Bergman space of the unit ball were considered by K. Zhu [Houston J. Math. 33, No. 1, 273–283 (2007; Zbl 1114.47031)]. His results could be extended to the case where the domain is the Harish-Chandra realization of an irreducible bounded symmetric domain in [X.-F. and Z.-J.Hu, Acta Math. Sci., Ser. B, Engl. Ed. 31, No. 2, 468–476 (2011; Zbl 1235.47026)].
Motivated by the above references, the author generalizes their works further to weighted Bergman spaces of a minimal bounded homogeneous domain and, using Zhu’s technique together with an integral formula, the author obtains the following theorem:
Assume that \(C_\varphi\) is a bounded operator on \(L_a^q ( {\mu ,dV_{\beta _0 } } )\) for some \( q > 0\) and \( \beta _0 > \varepsilon _{\min }\). Then \(C_\varphi\) is compact on \(L_a^p ( {\mu ,dV_{\beta } } )\) for any \( p > 0\) and \(\beta > \beta _0 + \varepsilon _\mu\) if and only if \[ \mathop {\lim }\limits_{z \to \partial \mu } \frac{{K_\mu ( {\varphi ( z ),\varphi ( z )} )}}{{K_\mu ( {z,z} )}} = 0. \]

MSC:

47B33 Linear composition operators
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
PDFBibTeX XMLCite
Full Text: arXiv Euclid

References:

[1] D. Békollé and A. T. Kagou, Reproducing properties and \(L^p\)-estimates for Bergman projections in Siegel domains of type II. , Studia. Math. 115 (1995), 219-239. · Zbl 0842.32016
[2] D. Békollé and C. Nana, \(L^{p}\)-boundedness of Bergman projections in the tube domain over Vinberg’s cone , J. Lie Theory 17 (2007), 115-144. · Zbl 1135.32004
[3] C. Cowen and B. MacCluer, Composition operators on spaces of analytic function , CRC Press, Boca Raton, 1994. · Zbl 0814.47040
[4] M. Engliš, Compact Toeplitz operators via the Berezin transform on bounded symmetric domains , Integr. Equ. Oper. Theory 20 (1999), 426-455. · Zbl 0936.47014 · doi:10.1007/BF01291836
[5] J. Fraut and A. Koranyi, Function spaces and reproducing kernels on bounded symmetric domains , J. Funct. Anal. 88 (1990), no.1, 64-89. · Zbl 0718.32026 · doi:10.1016/0022-1236(90)90119-6
[6] S. G. Gindikin, Analysis in homogeneous domains , Russian Math. Surveys 19 -4 (1964), 1-89. · Zbl 0144.08101
[7] H. Ishi and C. Kai, The representative domain of a homogeneous bounded domain , Kyushu J. Math. 64 (2010), 35-47. · Zbl 1195.32009 · doi:10.2206/kyushujm.64.35
[8] H. Ishi and S. Yamaji, Some estimates of the Bergman kernel of minimal bounded homogeneous domains , J. Lie Theory 21 (2011), 755-769. · Zbl 1235.32003
[9] X. Lv and Z. Hu, Compact composition operators on weighted Bergman spaces on bounded symmetric domains , Acta Math. Scientia 31B (2) (2011), 468-476. · Zbl 1235.47026 · doi:10.1016/S0252-9602(11)60247-6
[10] M. Maschler, Minimal domains and their Bergman kernel function , Pacific J. Math. 6 (1956), 501-516. · Zbl 0072.29902 · doi:10.2140/pjm.1956.6.501
[11] È. B. Vinberg, Homogeneous cones , Soviet Math. Dokl. 1 (1960), 787-790. · Zbl 0143.05203
[12] È. B. Vinberg, S. G. Gindikin, I. I. Pjateckiĭ-Šapiro, Classification and canonical realization of complex bounded homogeneous domains , Trans. Moscow Math. Soc. 12 (1963) 404-437. · Zbl 0137.05603
[13] S. Yamaji, Positive Toeplitz operators on the Bergman space of a minimal bounded homogeneous domain , to appear in Hokkaido Math. J. · Zbl 1248.47032
[14] K. H. Zhu, Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains , J. Oper. Theory 20 (1988), 329-357. · Zbl 0676.47016
[15] K. H. Zhu, Operator theory in function spaces, second edition , Amer. Math. Soc., Mathematical Surveys and Monographs Vol. 138 , 2007. · Zbl 1123.47001
[16] K. H. Zhu, Compact composition operators on weighted Bergman spaces of the unit ball , Houston J. Math. 33 (2007), 273-283. · Zbl 1114.47031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.