The Stević-Sharma operator on the Lipschitz space into the logarithmic Bloch space

Year 2023, Volume: 52 Issue: 3, 585 - 595, 30.05.2023

Hamid Vaezi , Soran Mahmoudfakheh

https://doi.org/10.15672/hujms.1104784

Abstract

In this paper, we study the boundedness and compactness of the Stević-Sharma operator on the Lipschitz space into the logarithmic Bloch space. Also, we give an estimate for the essential norm of the above operator.

Keywords

Stević-Sharma operator, Lipschtiz space, logarithmic Bloch space, weighted composition operator.

References

• [1] E. Abbasi and H. Vaezi, Estimates of essential norm of generalized weighted composition operators from Bloch type spaces to nth weighted-type spaces, Math. Slovaca 70 (1), 71-80, 2020.
• [2] E. Abbasi, H. Vaezi and S. Li, Essential norm of weighted composition operators from $H^{\infty}$ to $n$th weighted-type spaces, Mediterr. J. Math. 16, Article no:133, 2019.
• [3] J. Arazy, Multipliers of Bloch functions, University of Haifa Mathematics Publications Series 54, 1982.
• [4] K.R.M. Attele, Toeplitz and Hankel operators on Bergman one space, Hokkaido Math. J. 21 (2), 279-293, 1992.
• [5] R.E. Castillo, D.D. Clahane, J.F.F. Lopez and J.C.R. Fernndez, Composition operators from logarithmic Bloch spaces to weighted Bloch spaces, Appl. Math. Comput. 219 (12), 6692-6706, 2013.
• [6] F. Colonna and S. Li, Weighted composition operators from the Lipschitz space into the Zygmund space, MIA, Math. Inequal. Appl., Article ID 48478, 2007.
• [7] C.C. Cowen and B. MacCluer, Composition operators on spaces of analytic functions, Stud. Adv. Math., CRC Press, Boca Raton, 1995.
• [8] P.L. Duren, Theory of $H^{p}$ spaces, Academic press, New York, 1970.
• [9] P.L. Duren and B. W. Romberg, Shileds, A.L., Linear functionals on $H^{p}$ spaces with $0 < p < 1$. J. Reine Angew. Math. 238, 32-60, 1969.
• [10] P. Galanopoulos, On blog to $Q_{p}$ log pullbacks, J. Math. Anal. Appl. 337 (1), 712-725, 2008.
• [11] A.J. Garcia Ortiz and J.C. Ramos-Fernàndez, Composition operators from logarithmic Bloch spaces to Bloch-type spaces, Georgian Math. J. 20 (4), 671-686, 2013.
• [12] G.H. Hardy and J.E. Littlewood, Some properties of fractional integrals II, Math. Z. 34, 403-439, 1932.
• [13] M. Hassanlou, H. Vaezi and M. Wang, weighted composition operators on weak valued Bergman spaces and Hardy spaces, Banach J. Math. Anal. 9 (2), 35-43, 2015.
• [14] S. Li, E. Abbasi and H. Vaezi, Weighted composition operators from Bloch-type spaces to nth weighted-type spacese, Ann. Polon. Math. 124 (1), 93-107, 2020.
• [15] B. Maccluer and R. Zhao, Essential norm of weighted composition operators between Bloch-type spaces, Rocky M. J. Math. 33 (4), 1437-1458, 2003.
• [16] A. Petrov, Reverse estimates in logarithmic Bloch spaces, Arch. Math. (Basel) 100 (6), 551-560, 2013.
• [17] J. Shapiro, Composition Operators and alassical function theory, Springer-Verlag, New York, 1993.
• [18] S. Stević, A.K. Sharma and A. Bhat, Products of multiplication, composition and differentiation operators on weighted Bergman space, Appl. Math. Comput. 217 (20), 8115-8125, 2011.
• [19] M. Tjani, Compact composition operators on some Mobius invariant Banach spaces, Ph.D. thesis, Michigan State University, 1996.
• [20] H. Vaezi and S. Houdfar, Weighted composition operators between Besov-type spaces, Hacet. J. Math. Stat. 49 (1), 78-86, 2020.
• [21] H. Vaezi and S. Houdfar, Composition and weighted composition operators from Blochtype to Besov-type spaces, Math. Reports 22 (3-4), 297-308, 2020.
• [22] S. Ye, Multipliers and cyclic vectors on the weighted Bloch type space, Math. J. Okayama Univ. 48, 135-143, 2006
• [23] R. Yoneda, The composition operators on weighted Bloch space, Arch. Math. (Basel) 78 (4), 310-317, 2002.
• [24] F. Zhang and Y. Liu, Products of multiplication, composition and differentiation operators from mixed-norm spaces to weighted-type spaces, Taiwan. J. Math. 18 (6), 1927-1940, 2014.
• [25] K. Zhu, Bloch type spaces of analytic functions, Rocky Mountain J. Math. 23, 1143- 1177, 1993.