×

Compact weighted composition operators and multiplication operators between Hardy spaces. (English) Zbl 1167.47020

For \(N\in \mathbb N,\) let \(B_N\) be the unit ball of \(\mathbb C^N\). For each \(p\), \(1\leq p < \infty,\) the Hardy space \(H^p(B_N)\) consists of the holomorphic functions \(f\) in the ball such that
\[ \sup_{0<r<1}\int_{\partial B_N}|f(r\zeta)|^p\,d\sigma(\zeta)= \int_{\partial B_N}|f^*(\zeta)|^p\,d\sigma(\zeta)=\|f\|^p <\infty, \]
where \(d\sigma\) is the normalized Lebesgue measure on the boundary of \(B_N\) and \(f^*\) is the radial limit of \(f\), which exists for almost every \(\zeta \in \partial B_N\).
Given a holomorphic self-map \(\varphi\) of \(B_N\) and a holomorphic map \(u\) in \(B_N,\) the weighted composition operator \(M_uC_{\varphi}\) is defined by \(M_uC_{\varphi}f=u(f\circ\varphi),\) where \(f\) is holomorphic.
Let \(X\) and \(Y\) be Banach spaces and \(T\) be a bounded linear operator from \(X\) into \(Y\). The essential norm \(\|T\|_{e,X \to Y}\) is the distance from \(T\) to the set of compact operators from \(X\) into \(Y\).
The pull-back measure \(\mu_{\varphi,u}\) induced by the self-map \(\varphi\) and \(u\in H^q(B_N)\) is the finite positive Borel measure on \(\overline B_N\) defined by
\[ \mu_{\varphi,u}(E)=\int_{\varphi^{*-1}(E)}|u^*|^q\,d\sigma, \]
for all Borel sets \(E\). Notice that \(\varphi^*\) is a map of \(\partial B_N\) into \(\overline{B_N}\). For each \(\zeta \in \partial B_N\) and \(t>0,\) let the Carleson \(S(\zeta,t)\) be \(\{z \in \overline{B_N}:|1-\langle z,\zeta \rangle|<1\}\).
Many authors have studied weighted composition operators on different holomorphic function spaces. M. D. Contreras and A. Hernandez–Díaz [J. Math. Anal. Appl. 263, No. 1, 224–233 (2001; Zbl 1026.47016); Integral Equations Oper. Theory 46, No. 2, 165–188 (2003; Zbl 1042.47017)] characterized the compactness of \(M_uC_{\varphi}\) from \(H^p(B_1)\) into \(H^q(B_1)\) with \(1<p\leq q<\infty\) in terms of the pull-back measure, but they didn’t estimate \(\|M_uC_{\varphi}\|_{e, H^p\to H^q}\).
The authors’ main result is as follows. Let \(1<p\leq q<\infty\). If \(M_uC_{\varphi}\) is a bounded weighted composition operator from \(H^p(B_N)\) into \(H^q(B_N),\) then
\[ \begin{aligned} \| M_uC_{\varphi}\|_{e, H^p\to H^q}&\thicksim \limsup _{|w|\to 1^-}\int_{\partial B_N} |u^*(\zeta)|^q\left(\frac{1-|w|^2}{|1-\langle\varphi^*(\zeta),w\rangle|} \right)^{qN/p}d\sigma(\zeta)\\ &\thicksim \limsup_{t \to 0}\sup_{\zeta \in \partial B_N}\frac{\mu_{\varphi,u} (S(\zeta,t))}{t^{qN/p}}.\end{aligned} \]
The notation \(\thicksim\) means that the ratio of two terms are bounded below and above by constants dependent on the dimension \(N\) and other parameters.
They also show that a multiplication operator \(M_u\) from \(H^p(B_N)\) into \(H^q(B_N)\) with \(1<p\leq q<\infty\) is compact if and only if \(u=0\).

MSC:

47B33 Linear composition operators
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] B. R. Choe, “The essential norms of composition operators,” Glasgow Mathematical Journal, vol. 34, no. 2, pp. 143-155, 1992. · Zbl 0873.47017
[2] P. Gorkin and B. D. MacCluer, “Essential norms of composition operators,” Integral Equations and Operator Theory, vol. 48, no. 1, pp. 27-40, 2004. · Zbl 1065.47027
[3] P. Poggi-Corradini, “The essential norm of composition operators revisited,” in Studies on Composition Operators, vol. 213 of Contemporary Mathematics, pp. 167-173, American Mathematical Society, Providence, RI, USA, 1998. · Zbl 0873.47017
[4] J. H. Shapiro, “The essential norm of a composition operator,” Annals of Mathematics, vol. 125, no. 2, pp. 375-404, 1987. · Zbl 0642.47027
[5] S. Li and S. Stević, “Weighted composition operators between H\infty and \alpha -Bloch space in the unit ball,” to appear in Taiwanese Journal of Mathematics. · Zbl 0873.47017
[6] S. Li and S. Stević, “Weighted composition operators from H\infty to the Bloch space on the polydisc,” Abstract and Applied Analysis, vol. 2007, Article ID 48478, 13 pages pages, 2007. · Zbl 1152.47016
[7] G. Mirzakarimi and K. Seddighi, “Weighted composition operators on Bergman and Dirichlet spaces,” Georgian Mathematical Journal, vol. 4, no. 4, pp. 373-383, 1997. · Zbl 0891.47018
[8] S. Ohno, “Weighted composition operators between H\infty and the Bloch space,” Taiwanese Journal of Mathematics, vol. 5, no. 3, pp. 555-563, 2001. · Zbl 0873.47017
[9] S. Ohno, K. Stroethoff, and R. Zhao, “Weighted composition operators between Bloch-type spaces,” The Rocky Mountain Journal of Mathematics, vol. 33, no. 1, pp. 191-215, 2003. · Zbl 1042.47018
[10] S. Ohno and R. Zhao, “Weighted composition operators on the Bloch space,” Bulletin of the Australian Mathematical Society, vol. 63, no. 2, pp. 177-185, 2001. · Zbl 0873.47017
[11] M. D. Contreras and A. G. Hernández-Díaz, “Weighted composition operators on Hardy spaces,” Journal of Mathematical Analysis and Applications, vol. 263, no. 1, pp. 224-233, 2001. · Zbl 1026.47016
[12] M. D. Contreras and A. G. Hernández-Díaz, “Weighted composition operators between different Hardy spaces,” Integral Equations and Operator Theory, vol. 46, no. 2, pp. 165-188, 2003. · Zbl 0873.47017
[13] \breve Z. \breve Cu\breve cković and R. Zhao, “Weighted composition operators on the Bergman space,” Journal of the London Mathematical Society, vol. 70, no. 2, pp. 499-511, 2004. · Zbl 1069.47023
[14] \breve Z. \breve Cu\breve cković and R. Zhao, “Weighted composition operators between different weighted Bergman spaces and different Hardy spaces,” Illinois Journal of Mathematics, vol. 51, no. 2, pp. 479-498, 2007. · Zbl 0873.47017
[15] S. Ueki, “Weighted composition operators between weighted Bergman spaces in the unit ball of \Bbb Cn,” Nihonkai Mathematical Journal, vol. 16, no. 1, pp. 31-48, 2005. · Zbl 0873.47017
[16] L. Luo, “The essential norm of a composition operator on Hardy space of the unit ball,” Chinese Annals of Mathematics (Series A, Chinese), vol. 28A, no. 6, pp. 805-810, 2007, preprint. · Zbl 0873.47017
[17] W. Rudin, Function Theory in the Unit Ball of \Bbb Cn, vol. 241 of Fundamental Principles of Mathematical Science, Springer, New York, NY, USA, 1980. · Zbl 0873.47017
[18] S. C. Power, “Hörmander/s Carleson theorem for the ball,” Glasgow Mathematical Journal, vol. 26, no. 1, pp. 13-17, 1985. · Zbl 0873.47017
[19] B. D. MacCluer, “Compact composition operators on hp(bn),” The Michigan Mathematical Journal, vol. 32, no. 2, pp. 237-248, 1985. · Zbl 0585.47022
[20] K. H. Zhu, “Duality of Bloch spaces and norm convergence of Taylor series,” The Michigan Mathematical Journal, vol. 38, no. 1, pp. 89-101, 1991. · Zbl 0728.30026
[21] C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1995. · Zbl 0873.47017
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.