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Boundedness and essential norm of an integral-type operator on a Hilbert-Bergman-type spaces. (English) Zbl 1499.47021

Summary: Let \({\mathbb{D}}\) be the open unit disk of the complex plane \({\mathbb{C}}\) and \(H({\mathbb{D}} )\) be the space of all analytic functions on \({\mathbb{D}}\). Let \(A^2_{\gamma ,\delta }(\mathbb {D})\) be the space of analytic functions that are \(L^2\) with respect to the weight \(\omega_{\gamma ,\delta }(z)=( \ln \frac{1}{|z|})^{\gamma }[\ln (1-\frac{1}{\ln |z|})]^{\delta }\), where \(-1<\gamma <\infty\) and \(\delta \le 0\). For given \(g\in H({\mathbb{D}} )\), the integral-type operator \(I_g\) on \(H({\mathbb{D}} )\) is defined as \[I_gf(z)= \int_0^zf(\zeta )g(\zeta )\,d\zeta\,.\] In this paper, we characterize the boundedness of \(I_g\) on \(A^2_{\gamma ,\delta }\), whereas in the main result we estimate the essential norm of the operator. Some basic results on the space \(A^2_{\gamma ,\delta }({\mathbb{D}} )\) are also presented.

MSC:

47G10 Integral operators
45P05 Integral operators
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[1] Bierstedt, K.D., Summers, W.H.: Biduals of weighted Banach spaces of analytic functions. J. Aust. Math. Soc. Ser. A 54, 70-79 (1993) · Zbl 0801.46021 · doi:10.1017/S1446788700036983
[2] Kwon, E.G., Lee, J.: Composition operators between weighted Bergman spaces of logarithmic weights. Int. J. Math. 26(9), Article ID 1550068 (2015) · Zbl 1331.47042 · doi:10.1142/S0129167X15500688
[3] Pommerenke, C.: Schlichte funktionen und analytische funktionen von beschränkter mittlerer oszillation. Comment. Math. Helv. 52, 591-602 (1977) (in German) · Zbl 0369.30012 · doi:10.1007/BF02567392
[4] Chang, D.C., Li, S., Stević, S.: On some integral operators on the unit polydisk and the unit ball. Taiwan. J. Math. 11(5), 1251-1286 (2007) · Zbl 1149.47026 · doi:10.11650/twjm/1500404862
[5] Li, S., Stević, S.: Products of integral-type operators and composition operators between Bloch-type spaces. J. Math. Anal. Appl. 349, 596-610 (2009) · Zbl 1155.47036 · doi:10.1016/j.jmaa.2008.09.014
[6] Stević, S.: On a new operator from \(H∞H^{\infty }\) to the Bloch-type space on the unit ball. Util. Math. 77, 257-263 (2008) · Zbl 1175.47034
[7] Stević, S.: On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball. J. Math. Anal. Appl. 354, 426-434 (2009) · Zbl 1171.47028 · doi:10.1016/j.jmaa.2008.12.059
[8] Stević, S.: Boundedness and compactness of an integral operator on a weighted space on the polydisc. Indian J. Pure Appl. Math. 37(6), 343-355 (2006) · Zbl 1121.47032
[9] Sehba, B., Stević, S.: On some product-type operators from Hardy-Orlicz and Bergman-Orlicz spaces to weighted-type spaces. Appl. Math. Comput. 233, 565-581 (2014) · Zbl 1334.42052
[10] Zhu, X.: Generalized composition operators from generalized weighted Bergman spaces to Bloch type spaces. J. Korean Math. Soc. 46(6), 1219-1232 (2009) · Zbl 1197.47042 · doi:10.4134/JKMS.2009.46.6.1219
[11] Zhu, X.: Volterra composition operators from weighted-type spaces to Bloch-type spaces and mixed norm spaces. Math. Inequal. Appl. 14(1), 223-233 (2011) · Zbl 1242.47021
[12] Du, J., Li, S., Zhang, Y.: Essential norm of weighted composition operators on Zygmund-type spaces with normal weight. Math. Inequal. Appl. 21(3), 701-714 (2018) · Zbl 06948271
[13] Hu, Q., Shi, Y.F., Shi, Y.C., Zhu, X.: Essential norm of generalized weighted composition operators from the Bloch space to the Zygmund space. J. Inequal. Appl. 2016, Article ID 123 (2016) · Zbl 1337.30066 · doi:10.1186/s13660-016-1066-4
[14] Shi, Y., Li, S.: Essential norm of integral operators on Morrey type spaces. Math. Inequal. Appl. 19(1), 385-393 (2016) · Zbl 1335.30020
[15] Stević, S.: Norm and essential norm of composition followed by differentiation from α-Bloch spaces to Hμ∞\(H^{\infty }_{\mu }\). Appl. Math. Comput. 207, 225-229 (2009) · Zbl 1157.47026
[16] Stević, S., Sharma, A.K., 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
[17] Zhu, X.: Essential norm of generalized weighted composition operators on Bloch-type spaces. Appl. Math. Comput. 274, 133-142 (2016) · Zbl 1410.30032
[18] Kwon, E.G., Lee, J.: Essential norm of the composition operators between Bergman spaces of logarithmic weights. Bull. Korean Math. Soc. 54(1), 187-198 (2017) · Zbl 06699734 · doi:10.4134/BKMS.b160014
[19] Aleman, A., Siskakis, A.: Integration operators on Bergman spaces. Indiana Univ. Math. J. 46, 337-356 (1997) · Zbl 0951.47039 · doi:10.1512/iumj.1997.46.1373
[20] Li, S., Stević, S.: Riemann-Stieltjes operators between different weighted Bergman spaces. Bull. Belg. Math. Soc. Simon Stevin 15(4), 677-686 (2008) · Zbl 1169.47026
[21] Jiang, Z.J.: Product-type operators from logarithmic Bergman-type spaces to Zygmund-Orlicz spaces. Mediterr. J. Math. 13(6), 4639-4659 (2016) · Zbl 1356.47038 · doi:10.1007/s00009-016-0767-8
[22] Tjani, M.: Compact composition operators on some Möbius invariant Banach space. PhD thesis, Michigan State University (1996)
[23] Nehari, Z.: Conformal Mapping. Dover, New York (1975) · Zbl 0048.31503
[24] Shabat, B.V.: Introduction to Complex Analysis Part II Functions of Several Variables. Am. Math. Soc., Providence (1992) · Zbl 0799.32001 · doi:10.1090/mmono/110
[25] Stević, S., Ueki, S.I.: Weighted composition operators from the weighted Bergman space to the weighted Hardy space on the unit ball. Appl. Math. Comput. 215, 3526-3533 (2010) · Zbl 1197.47040
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