×

Differences of Stević-Sharma operators. (English) Zbl 1508.47088

Summary: A generalization of the products of composition, multiplication and differentiation operators is the Stević-Sharma operator \(T_{u_1,u_2,\varphi}\), defined by \(T_{u_1,u_2,\varphi}f=u_1\cdot f\circ\varphi+u_2\cdot f'\circ\varphi\), where \(u_1,u_2,\varphi\) are holomorphic functions on the unit disk \(\mathbb{D}\) in the complex plane \(\mathbb{C}\) and \(\varphi(\mathbb{D})\subset\mathbb{D}\). We are interested in the difference of Stević-Sharma operators which has never been considered so far. In this paper, we characterize its boundedness, compactness and order boundedness between Banach spaces of holomorphic functions. As an important special case, we obtain the above characterizations of the difference of weighted composition operators. Furthermore, we show the equivalence of order boundedness and Hilbert-Schmidtness for the difference of composition operators between Hardy or weighted Bergman spaces.

MSC:

47B91 Operators on complex function spaces
47B33 Linear composition operators
46E15 Banach spaces of continuous, differentiable or analytic functions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Acharyya, S.; Ferguson, T., Sums of weighted differentiation composition operators, Complex Anal. Oper. Theory, 13, 1465-1479 (2019) · Zbl 1480.47049
[2] Acharyya, S.; Wu, Z., Compact and Hilbert-Schmidt differences of weighted composition operators, Integral Equ. Oper Theory, 88, 465-482 (2017) · Zbl 1466.47019
[3] Bear, H., Lectures on Gleason Parts. Lecture Notes in Mathematics (1970), Berlin-New York: Springer-Verlag, Berlin-New York · Zbl 0203.44601
[4] Choe, B.; Hosokawa, T.; Koo, H., Hilbert-Schmidt differences of composition operators on the Bergman space, Math. Z., 269, 751-775 (2011) · Zbl 1234.47009
[5] Choe, B.; Koo, H.; Wang, M.; Yang, J., Compact linear combinations of composition operators induced by linear fractional maps, Math. Z., 280, 807-824 (2015) · Zbl 1326.47022
[6] Choe, B.; Koo, H.; Wang, M., Compact double differences of composition operators on the Bergman spaces, J. Funct. Anal., 272, 2273-2307 (2017) · Zbl 1437.47010
[7] Contreras, M.; Hernández-Díaz, A., Weighted composition operators on spaces of functions with derivative in a Hardy space, J. Oper. Thoery, 52, 173-184 (2004) · Zbl 1104.47027
[8] Cowen, C.; MacCluer, B., Composition operators on spaces of analytic functions (1995), Boca Raton: CRC Press, Boca Raton · Zbl 0873.47017
[9] Diestel, J.; Jarchow, H.; Tonge, A., Absolutely summing operators (1995), Cambridge: Cambridge University Press, Cambridge · Zbl 0855.47016
[10] Galanopoulos, P.; Girela, D.; Peláez, J.; Siskakis, A., Generalized Hilbert operators, Ann. Acad. Sci. Fenn. Math., 39, 231-258 (2014) · Zbl 1297.47030
[11] Girela, D.; Peláez, J., Carleson measures, multipliers and integration operators for spaces of Dirichlet type, J. Funct. Anal., 241, 334-358 (2006) · Zbl 1115.46020
[12] Guo, X.; Wang, M., Difference of weighted composition operators on the space of Cauchy integral transforms, Taiwan. J. Math., 22, 1435-1450 (2018) · Zbl 1473.47005
[13] Hai, P.; Putinar, M., Complex symmetric differential operators on Fock space, J. Differ. Equ., 265, 4213-4250 (2018) · Zbl 1408.30033
[14] Hedenmalm, H.; Korenblum, B.; Zhu, K., Theory of Bergman Spaces (2000), New York: Springer, New York · Zbl 0955.32003
[15] Hosokawa, T., Differences of weighted composition operators on the Bloch spaces, Complex Anal. Oper. Theory, 3, 847-866 (2009) · Zbl 1216.47037
[16] Hosokawa, T.; Ohno, S., Differences of composition operators on the Bloch spaces, J. Oper. Theory, 57, 229-242 (2007) · Zbl 1174.47019
[17] Hosokawa, T.; Ohno, S., Differences of weighted composition operators from \(H^\infty\) to Bloch space, Taiwan. J. Math., 16, 2093-2105 (2012) · Zbl 1283.47038
[18] Hu, Q.; Li, S.; Shi, Y., A new characterization of differences of weighted composition operators on weighted-type spaces, Comput. Method Funct. Theory, 17, 303-318 (2017) · Zbl 1432.30042
[19] Hunziker, H.; Jarchow, H., Composition operators which improve integrability, Math. Nachr., 152, 83-99 (1991) · Zbl 0760.47015
[20] Jiang, Z.; Stević, S., Compact differences of weighted composition operators from weighted Bergman spaces to weighted-type spaces, Appl. Math. Comput., 217, 3522-3530 (2010) · Zbl 1204.30043
[21] Koo, H.; Wang, M., Joint Carleson measure and the difference of composition operators on \(A^p_\alpha ({\mathbb{B}}_n)\), J. Math. Anal. Appl., 419, 1119-1142 (2014) · Zbl 1294.47038
[22] Koo, H.; Wang, M., Cancellation properties of composition operators on Bergman spaces, J. Math. Anal. Appl., 432, 1174-1182 (2015) · Zbl 1321.47062
[23] Lin, Q.; Liu, J.; Wu, Y., Volterra type operators on \(S^p({\mathbb{D}})\) spaces, J. Math. Anal. Appl., 461, 1100-1114 (2018) · Zbl 06852149
[24] Liu, Y.; Yu, Y., Products of composition, multiplication and radial derivative operators from logarithmic Bloch spaces to weighted-type spaces on the unit ball, J. Math. Anal. Appl., 423, 76-93 (2015) · Zbl 1304.47046
[25] Liu, Y.; Yu, Y., On an extension of Stević-Sharma operator from the general space to weighted-type spaces on the unit ball, Complex Anal. Oper. Theory, 11, 261-288 (2017) · Zbl 1375.47029
[26] Moorhouse, J., Compact differences of composition operators, J. Funct. Anal., 219, 70-92 (2005) · Zbl 1087.47032
[27] Nieminen, P., Compact differences of composition operators on Bloch and Lipschitz spaces, Comput. Method Funct. Theory, 7, 325-344 (2007) · Zbl 1146.47016
[28] Peláez, J.; Pérez-González, F.; Rättyä, J., Operator theoretic differences between Hardy and Dirichlet-type spaces, J. Math. Anal. Appl., 418, 387-402 (2014) · Zbl 1310.30047
[29] Saukko, E., An application of atomic decomposition in Bergman spaces to the study of differences of composition operators, J. Funct. Anal., 262, 3872-3890 (2012) · Zbl 1276.47032
[30] Sharma, A., On order bounded weighted composition operators between Dirichlet spaces, Positivity, 21, 1213-1221 (2017) · Zbl 1439.47024
[31] Sharma, M.; Sharma, A., On order bounded difference of weighted composition operators between Hardy spaces, Complex Anal. Oper. Theory, 13, 2191-2201 (2019) · Zbl 1479.47025
[32] Shi, Y.; Li, S., Differences of composition operators on Bloch type spaces, Complex Anal. Oper. Theory, 11, 227-242 (2017) · Zbl 1361.30098
[33] Stević, S.; Sharma, A.; Bhat, A., Products of multiplication composition and differentiation operators on weighted Bergman spaces, Appl. Math. Comput., 217, 8115-8125 (2011) · Zbl 1218.30152
[34] Stević, S.; Sharma, A.; Bhat, A., Essential norm of products of multiplication composition and differentiation operators on weighted Bergman spaces, Appl. Math. Comput., 218, 2386-2397 (2011) · Zbl 1244.30080
[35] Ueki, S., Order bounded weighted composition operators mapping into the Bergman space, Complex Anal. Oper. Theory, 6, 549-560 (2012) · Zbl 1283.47029
[36] Wang, S., Wang, M., Guo, X.: Products of composition, multiplication and iterated differentiation operators between Banach spaces of holomorphic functions, Taiwanese J. Math., 10.11650/tjm/190405 · Zbl 1444.47050
[37] Weidmann, J., Linear operators in Hilbert spaces (1980), New York-Berlin: Springer-Verlag, New York-Berlin · Zbl 0434.47001
[38] Yu, Y.; Liu, Y., On Stević type operator from \(H^\infty\) space to the logarithmic Bloch spaces, Complex Anal. Oper. Theory, 9, 1759-1780 (2015) · Zbl 1357.47031
[39] Zhang, F.; Liu, Y., On a Stević-Sharma operator from Hardy spaces to Zygmund-type spaces on the unit disk, Complex Anal. Oper. Theory, 12, 81-100 (2018) · Zbl 06838035
[40] Zhang, L.; Zhou, Z., Hilbert-Schmidt differences of composition operators between the weighted Bergman spaces on the unit ball, Banach J. Math. Anal., 7, 160-172 (2013) · Zbl 1281.47014
[41] Zhao, R., Essential norms of composition operators between Bloch type spaces, Proc. Amer. Math. Soc., 138, 2537-2546 (2010) · Zbl 1190.47028
[42] Zhu, K., Bloch type spaces of analytic functions, Rocky Mountain J. Math., 23, 1143-1177 (1993) · Zbl 0787.30019
[43] Zhu, K., Operator Theory in Function Spaces (2007), Providence: American Mathematical Society, Providence · Zbl 1123.47001
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.