Composition operators on the normal weight Dirichlet type space in the unit disc

Abstract

Let p>0 and \nu be a normal function on [0,1). In this paper, several equivalent characterizations are given for which composition operators are bounded or compact on the normal weight Dirichlet type space {\mathcal{D}}_{\nu }^p \left(D\right) in the unit disc.

Keywords

Composition operator / normal weight Dirichlet type space / boundedness / compactness / unit disc

References

[1] Abdollahi A. Self-commutators of automorphic composition operators on the Dirichlet space. Proc Amer Math Soc 2008; 136(9): 3185–3193

[2] Cao Y X, Kumar S, Zhou Z H. Order bounded weighted composition operators mapping into the Dirichlet type space. Chin Ann Math Ser B 2016; 37(4): 585–594

[3] Cheng Y, Kumar S, Zhou Z H. Composition operators on Dirichlet spaces and Bloch space. Acta Math Sin (Engl Ser) 2014; 30(10): 1775–1784

[4] CowenCMacCluer B. Composition operators on spaces of analytic functions. Boca Raton: CRC Press, 1995

[5] Hammond C. The norm of a composition operator with linear symbol acting on the Dirichlet space. J Math Anal Appl 2005; 303(2): 499–508

[6] Kumar R, Singh K. Essential normal of weighted composition operators on the Dirichlet space. Extracta Math 2006; 21: 249–259

[7] Kumar S. Weighted composition operators between spaces of Dirichlet type. Rev Mat Complut 2009; 22: 469–488

[8] Lefèvre P, Li D, Queffélec H, Rodríguez-Piazza L. Compact composition operators on the Dirichlet space and capacity of sets of contact points. J Funct Anal 2013; 264(4): 895–919

[9] Li D, Queffélec H, Rodríguez-Piazza L. Two results on composition operators on the Dirichlet space. J Math Anal Appl 2015; 426(2): 734–746

[10] Liu X S, Lou Z J. Weighted composition operators on weighted Dirichlet spaces. Acta Math Sci Ser B 2013; 33(4): 1119–1126

[11] Luecking D. Bounded composition operators with closed range on the Dirichlet space. Proc Amer Math Soc 2000; 128(4): 1109–1116

[12] Martin M, Vukotić D. Norms and spectral radii of composition operators acting on the Dirichlet space. J Math Anal Appl 2005; 304: 22–32

[13] Mirzakarimi G, Seddighi K. Weighted composition operators on Bergman and Dirichlet spaces. Georgian Math J 1997; 4: 373–383

[14] Patton L, Robbins M. Approximating composition operator norms on the Dirichlet space. J Math Anal Appl 2008; 339(2): 1374–1385

[15] RudinW. Function theory in the unit ball of Cⁿ. New York: Springer-Verlag, 1980

[16] ShapiroJ. Composition operators and classical function theory. New York: Springer-verlag, 1993

[17] Wang M F. Weighted composition operators between Dirichlet spaces. Acta Math Sci Ser B 2011; 31(2): 641–651

[18] Zhang L, Zhou Z H. Hypercyclicity of weighted composition operators on a weighted Dirichlet space. Complex Var Elliptic Equ 2014; 59(7): 1043–1051

[19] Zhang X J, Xi L H, Fan H X, Li J F. Atomic decomposition of μ-Bergman space in Cⁿ. Acta Math Sci Ser B 2014; 34(3): 779–789

[20] Zorboska N. Composition operators on weighted Dirichlet spaces. Proc Amer Math Soc 1998; 126: 2013–2023

[21] ZhuK. Spaces of holomorphic functions in the unit ball. New York: Springer-Verlag, 2005

[22] ZhuK. Operator theory in function spaces. 2nd ed. Math Surveys Monogr, Vol 138. Providence, RI: AMS, 2007

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Higher Education Press 2022