Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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DIFFERENCES OF DIFFERENTIAL OPERATORS BETWEEN WEIGHTED-TYPE SPACES

  • Received : 2020.03.17
  • Accepted : 2021.02.03
  • Published : 2021.07.31
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Abstract

Let 𝓗(𝔻) be the space of analytic functions on the unit disc 𝔻. Let 𝜓 = (𝜓j)nj=0 and 𝚽 = (𝚽j)nj=0 be such that 𝜓j, 𝚽j ∈ 𝓗(𝔻). The linear differential operator is defined by T𝜓(f) = ∑nj=0 𝜓jf(j), f ∈ 𝓗(𝔻). We characterize the boundedness and compactness of the difference operator (T𝜓 - T𝚽)(f) = ∑nj=0 (𝜓j - 𝚽j) f(j) between weighted-type spaces of analytic functions. As applications, we obtained boundedness and compactness of the difference of multiplication operators between weighted-type and Bloch-type spaces. Also, we give examples of unbounded (non compact) differential operators such that their difference is bounded (compact).

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References

  1. E. Berkson, Composition operators isolated in the uniform operator topology, Proc. Amer. Math. Soc. 81 (1981), no. 2, 230-232. https://doi.org/10.2307/2044200
  2. K. D. Bierstedt, J. Bonet, and A. Galbis, Weighted spaces of holomorphic functions on balanced domains, Michigan Math. J. 40 (1993), no. 2, 271-297. https://doi.org/10.1307/mmj/1029004753
  3. K. D. Bierstedt, J. Bonet, and J. Taskinen, Associated weights and spaces of holomorphic functions, Studia Math. 127 (1998), no. 2, 137-168. https://doi.org/10.4064/sm-127-2-137-168
  4. K. D. Bierstedt and W. H. Summers, Biduals of weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Ser. A 54 (1993), no. 1, 70-79. https://doi.org/10.1017/S1446788700036983
  5. J. Bonet, P. Doma'nski, M. Lindstrom, and J. Taskinen, Composition operators between weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Ser. A 64 (1998), no. 1, 101-118. https://doi.org/10.1017/S1446788700001336
  6. C. Chen and Z.-H. Zhou, Differences of the weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces, Ukrainian Math. J. 68 (2016), no. 6, 959-971; translated from Ukrain. Mat. Zh. 68 (2016), no. 6, 842-852. https://doi.org/10.1007/s11253-016-1269-3
  7. C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.
  8. D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Oxford University Press, 2018.
  9. R. J. Fleming and J. E. Jamison, Isometries on Banach spaces: function spaces, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 129, Chapman & Hall/CRC, Boca Raton, FL, 2003.
  10. R. J. Fleming and J. E. Jamison, Isometries on Banach spaces. Vol. 2, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 138, Chapman & Hall/CRC, Boca Raton, FL, 2008.
  11. S. R. Garcia, E. Prodan, and M. Putinar, Mathematical and physical aspects of complex symmetric operators, J. Phys. A 47 (2014), no. 35, 353001, 54 pp. https://doi.org/10.1088/1751-8113/47/35/353001
  12. K. Guo and H. Huang, Multiplication operators on the Bergman space, Lecture Notes in Mathematics, 2145, Springer, Heidelberg, 2015. https://doi.org/10.1007/978-3-662-46845-6
  13. X. Guo and M. Wang, Difference of weighted composition operators on the space of Cauchy integral transforms, Taiwanese J. Math. 22 (2018), no. 6, 1435-1450. https://doi.org/10.11650/tjm/180404
  14. P. V. Hai and M. Putinar, Complex symmetric differential operators on Fock space, J. Differential Equations 265 (2018), no. 9, 4213-4250. https://doi.org/10.1016/j.jde.2018.06.003
  15. R. A. Hibschweiler and N. Portnoy, Composition followed by differentiation between Bergman and Hardy spaces, Rocky Mountain J. Math. 35 (2005), no. 3, 843-855. https://doi.org/10.1216/rmjm/1181069709
  16. T. Hosokawa, K. Izuchi, and S. Ohno, Topological structure of the space of weighted composition operators on H∞, Integral Equations Operator Theory 53 (2005), no. 4, 509-526. https://doi.org/10.1007/s00020-004-1337-1
  17. Z.-J. Jiang, Product-type operators from Zygmund spaces to Bloch-Orlicz spaces, Complex Var. Elliptic Equ. 62 (2017), no. 11, 1645-1664. https://doi.org/10.1080/17476933.2016.1278436
  18. X. Liu and S. Li, Differences of generalized weighted composition operators from the Bloch space into Bers-type spaces, Filomat 31 (2017), no. 6, 1671-1680. https://doi.org/10.2298/FIL1706671L
  19. W. Lusky, On the structure of Hv0(D) and hv0(D), Math. Nachr. 159 (1992), 279-289. https://doi.org/10.1002/mana.19921590119
  20. W. Lusky, On weighted spaces of harmonic and holomorphic functions, J. London Math. Soc. (2) 51 (1995), no. 2, 309-320. https://doi.org/10.1112/jlms/51.2.309
  21. B. MacCluer, S. Ohno, and R. Zhao, Topological structure of the space of composition operators on H∞, Integral Equations Operator Theory 40 (2001), no. 4, 481-494. https://doi.org/10.1007/BF01198142
  22. J. S. Manhas, Compact differences of weighted composition operators on weighted Banach spaces of analytic functions, Integral Equations Operator Theory 62 (2008), no. 3, 419-428. https://doi.org/10.1007/s00020-008-1630-5
  23. J. S. Manhas and R. Zhao, Products of weighted composition and differentiation operators into weighted Zygmund and Bloch spaces, Acta Math. Sci. Ser. B (Engl. Ed.) 38 (2018), no. 4, 1105-1120. https://doi.org/10.1016/S0252-9602(18)30802-6
  24. J. Moorhouse, Compact differences of composition operators, J. Funct. Anal. 219 (2005), no. 1, 70-92. https://doi.org/10.1016/j.jfa.2004.01.012
  25. S. Ohno, Products of composition and differentiation between Hardy spaces, Bull. Austral. Math. Soc. 73 (2006), no. 2, 235-243. https://doi.org/10.1017/S0004972700038818
  26. I. Park, Compact differences of composition operators on large weighted Bergman spaces, J. Math. Anal. Appl. 479 (2019), no. 2, 1715-1737. https://doi.org/10.1016/j.jmaa.2019.07.020
  27. M. Reed and B. Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press, New York, 1975.
  28. L. A. Rubel and A. L. Shields, The second duals of certain spaces of analytic functions, J. Austral. Math. Soc. 11 (1970), 276-280. https://doi.org/10.1017/S1446788700006649
  29. J. H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. https://doi.org/10.1007/978-1-4612-0887-7
  30. J. H. Shapiro and C. Sundberg, Isolation amongst the composition operators, Pacific J. Math. 145 (1990), no. 1, 117-152. http://projecteuclid.org/euclid.pjm/1102645610 https://doi.org/10.2140/pjm.1990.145.117
  31. R. K. Singh and J. S. Manhas, Composition operators on function spaces, North-Holland Mathematics Studies, 179, North-Holland Publishing Co., Amsterdam, 1993.
  32. X.-J. Song and Z.-H. Zhou, Differences of weighted composition operators from Bloch space to H∞ on the unit ball, J. Math. Anal. Appl. 401 (2013), no. 1, 447-457. https://doi.org/10.1016/j.jmaa.2012.12.030
  33. S. Stevic, Generalized composition operators between mixed-norm and some weighted spaces, Numer. Funct. Anal. Optim. 29 (2008), no. 7-8, 959-978. https://doi.org/10.1080/01630560802282276
  34. S. Stevic and Z. J. Jiang, 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. https://doi.org/10.11650/twjm/1500406489
  35. P. T. Tien and L. H. Khoi, Differences of weighted composition operators between the Fock spaces, Monatsh. Math. 188 (2019), no. 1, 183-193. https://doi.org/10.1007/s00605-018-1179-6
  36. S. Wang, M. Wang, and X. Guo, Differences of Stevi'c-Sharma operators, Banach J. Math. Anal. 14 (2020), no. 3, 1019-1054. https://doi.org/10.1007/s43037-019-00051-z
  37. M. Wang, X. Yao, and F. Chen, Compact differences of weighted composition operators on the weighted Bergman spaces, J. Inequal. Appl. 2017 (2017), Paper No. 2, 14 pp. https://doi.org/10.1186/s13660-016-1277-8
  38. E. Wolf, Differences of composition operators between weighted Bergman spaces and weighted Banach spaces of holomorphic functions, Glasg. Math. J. 52 (2010), no. 2, 325-332. https://doi.org/10.1017/S0017089510000029
  39. E. Wolf, Composition followed by differentiation between weighted Banach spaces of holomorphic functions, Rev. R. Acad. Cienc. Exactas F'is. Nat. Ser. A Mat. RACSAM 105 (2011), no. 2, 315-322. https://doi.org/10.1007/s13398-011-0040-8
  40. L. Xiaoman and Y. Yanyan, Product of differentiation operator and multiplication operator from H∞ to Zygmund spaces, J. Xuzhou Normal Univ. (Natural Science Edition) 29 (2011), no. 1, 37-39. https://doi.org/10.3969/j.issn.1007-6573.2011.01.009
  41. Y. Y. Yu and Y. M. Liu, The product of differentiation and multiplication operators from the mixed-norm to the Bloch-type spaces, Acta Math. Sci. Ser. A (Chin. Ed.) 32 (2012), no. 1, 68-79.
  42. K. Zhu, Operator theory in function spaces, second edition, Mathematical Surveys and Monographs, 138, American Mathematical Society, Providence, RI, 2007. https://doi.org/10.1090/surv/138
  43. X. Zhu, Products of differentiation, composition and multiplication from Bergman type spaces to Bers type spaces, Integral Transforms Spec. Funct. 18 (2007), no. 3-4, 223-231. https://doi.org/10.1080/10652460701210250