## Taiwanese Journal of Mathematics

### COMPACTNESS OF THE DIFFERENCES OF WEIGHTED COMPOSITION OPERATORS FROM WEIGHTED BERGMAN SPACES TO WEIGHTED-TYPE SPACES ON THE UNIT BALL

#### Abstract

Let $\varphi_{1}$ and $\varphi_{2}$ be holomorphic self-maps of the open unit ball $\mathbb B$ in $\mathbb C^N$, $u_{1}$ and $u_{2}$ be holomorphic functions on $\mathbb B$ and let weighted composition operators $W_{\varphi_{1},u_{1}}$; $W_{\varphi_{2},u_{2}}: A^{p}_{\alpha} \to H^{\infty}_{v}$ be bounded. This paper characterizes the compactness of the difference of these operators from the weighted Bergman space $A^p_\alpha$, $0-1$, to the weighted-type space $H^\infty_v$ of holomorphic functions on $\mathbb B$ in terms of inducing symbols $\varphi_{1}$, $\varphi_{2}$, $u_{1}$ and $u_{2}$. For the case $p \gt 1$ we find an asymptotically equivalent expression to the essential norm of the operator.

#### Article information

Source
Taiwanese J. Math., Volume 15, Number 6 (2011), 2647-2665.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406489

Digital Object Identifier
doi:10.11650/twjm/1500406489

Mathematical Reviews number (MathSciNet)
MR2896136

Zentralblatt MATH identifier
1315.47034

#### Citation

Stević, Stevo; Jiang, Zhi Jie. COMPACTNESS OF THE DIFFERENCES OF WEIGHTED COMPOSITION OPERATORS FROM WEIGHTED BERGMAN SPACES TO WEIGHTED-TYPE SPACES ON THE UNIT BALL. Taiwanese J. Math. 15 (2011), no. 6, 2647--2665. doi:10.11650/twjm/1500406489. https://projecteuclid.org/euclid.twjm/1500406489

#### References

• K. D. Bierstedt and W. H. Summers, Biduals of weighted Banach spaces of analytic functions, J. Austral. Math. Soc. $($Series A$)$, 54 (1993), 70-79.
• K. D. Bierstedt, J. Bonet and J. Taskinen, Associated weights and spaces of holomorphic functions, Studia Math., 127 (1998), 70-79.
• J. Bonet, P. Domanski and M. Lindström, Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions, Canad. Math. Bull., 42(2), (1999), 139-148.
• J. Bonet, M. Lindström and E. Wolf, Differences of composition operators between weighted Banach spaces of holomorphic functions, J. Austral. Math. Soc., 84 (2008), 9-20.
• D. Clahane and S. Stevi\' c, Norm equivalence and composition operators between Bloch/Lipschitz spaces of the unit ball, J. Inequal. Appl., Vol. 2006, Article ID 61018, (2006), 11 pages.
• M. D. Contreras and A. G. Hernández-Diaz, Weighted composition operators in weighted Banach spaces of analytic functions, J. Austral. Math. Soc. $($Serie A$)$, 69 (2000), 41-60.
• C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, 1995.
• T. Hosokawa and K. Izuchi, Essential norms of differences of composition operators on $H^{\infty}$, J. Math. Japan, 57 (2005), 669-690.
• T. Hosokawa, K. Izuchi and D. Zheng, Isolated points and essential components of composition operators on $H^{\infty}$, Proc. Amer. Math. Soc., 130 (2002), 1765-1773.
• T. Hosokawa and S. Ohno, Differences of composition operators on the Bloch spaces, J. Operator Theory, 57 (2007), 229-242.
• S. Li and S. Stević, Weighted composition operators from $\a$-Bloch space to $H^\infty$ on the polydisk, Numer. Funct. Anal. Optimization, 28(7) (2007), 911-925.
• S. Li and S. Stevi\' c, Weighted composition operators between $H^\infty$ and $\alpha$-Bloch spaces in the unit ball, Taiwanese J. Math., 12 (2008), 1625-1639.
• S. Li and S. Stevi\' c, Weighted composition operators from Zygmund spaces into Bloch spaces, Appl. Math. Comput., 206(2) (2008), 825-831.
• M. Lindström and E. Wolf, Essential norm of the difference of weighted composition operators, Monatsh. Math., 153 (2008), 133-143.
• W. Lusky, On weighted spaces of harmonic and holomorphic functions, J. London Math. Soc., 51 (1995), 309-320.
• B. D. MacCluer, S. Ohno and R. Zhao, Topological structure of the space of composition operators on $H^{\infty}$, Integral Equations Operator Theory, 40 (2001), 481-494.
• A. Montes-Rodriguez, Weighted composition operators on weighted Banach spaces of analytic functions. J. London Math. Soc., 61(3) (2000), 872-884.
• P. J. Nieminen, Compact differences of composition operators on Bloch and Lipschitz spaces, Comput. Methods Funct. Theory, 7(2) (2007), 325-344.
• S. Stević, Composition operators between $H^{\infty}$ and the $\alpha$-Bloch spaces on the polydisc, Z. Anal. Anwend., 25 (2006), 457-466.
• S. Stević, Weighted composition operators between mixed norm spaces and $H_{\alpha}^{\infty}$ spaces in the unit ball, J. Inequal. Appl., Vol. 2007, Article ID 28629, 2007, 9 pages.
• S. Stevi\' c, Generalized composition operators from logarithmic Bloch spaces to mixed-norm spaces, Util. Math., 77 (2008), 167-172.
• S. Stević, Norm of weighted composition operators from Bloch space to $H_{\mu}^{\infty}$ on the unit ball, Ars. Combin., 88 (2008), 125-127.
• S. Stević, Norms of some operators from Bergman spaces to weighted and Bloch-type space, Util. Math., 76 (2008), 59-64.
• S. Stevi\' c, On a new operator from $H^\infty$ to the Bloch-type space on the unit ball, Util. Math., 77 (2008), 257-263.
• S. Stevi\' c, Essential norm of an operator from the weighted Hilbert-Bergman space to the Bloch-type space, Ars Combin., 91 (2009), 123-127.
• S. Stevi\' c, Essential norms of weighted composition operators from the Bergman space to weighted-type spaces on the unit ball, Ars. Combin., 91 (2009), 391-400.
• S. Stevi\' c, Integral-type operators from the mixed-norm space to the Bloch-type space on the unit ball, Siberian Math. J., 50(6) (2009), 1098-1105.
• S. Stević, Norm and essential norm of composition followed by differentiation from $\alpha$-Bloch spaces to $H_{\mu}^{\infty}$, Appl. Math. Comput., 207 (2009), 225-229.
• S. Stevi\' c, Norm of weighted composition operators from $\a$-Bloch spaces to weighted-type spaces, Appl. Math. Comput., 215 (2009), 818-820.
• S. Stevi\' c, Norms of some operators on the Bergman and the Hardy space in the unit polydisk and the unit ball, Appl. Math. Comput., 215 (6) (2009), 2199-2205.
• S. Stevi\' c, On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball, J. Math. Anal. Appl., 354 (2009), 426-434.
• S. Stevi\' c, Products of integral-type operators and composition operators from the mixed norm space to Bloch-type spaces, Siberian Math. J., 50(4) (2009), 726-736.
• S. Stevi\' c, Weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball, Appl. Math. Comput., 212 (2009), 499-504.
• S. Stevi\' c, Weighted differentiation composition operators from $H^\infty$ and Bloch spaces to $n$th weigthed-type spaces on the unit disk, Appl. Math. Comput., 216 (2010), 3634-3641.
• S. Stevi\' c and E. Wolf, Differences of composition operators between weighted-type spaces of holomorphic functions on the unit ball of $\CC^n$, Appl. Math. Comput., 215(5) (2009), 1752-1760.
• S. I. Ueki, Weighted composition operator on the Fock space, Proc. Amer. Math. Soc., 135 (2007), 1405-1410.
• S. I. Ueki and L. Luo, Essential norms of weighted composition operators between weighted Bergman spaces of the ball, Acta Sci. Math. $($Szeged$)$, 74 (2008), 829-843.
• W. Yang, Weighted composition operators from Bloch-type spaces to weighted-type spaces, Ars Combin., 93 (2009), 265-274.
• K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer, New York, 2005.
• X. Zhu, Weighted composition operators from logarithmic Bloch spaces to a class of weighted-type spaces in the unit ball, Ars. Combin., 91 (2009), 87-95.