Abstract
We study weighted composition operators acting between Fock spaces. The following results are obtained:
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(i) Criteria for the boundedness and compactness.
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(ii) Characterizations of compact differences and essential norm.
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(iii) Complete descriptions of path connected components and isolated points of the space of composition operators and the space of nonzero weighted composition operators.
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Beltrán-Meneu, M.J., Gómez-Collado, M.C., Jordá, E., Jornet, D.: Mean ergodicity of weighted composition operators on spaces of holomorphic functions. J. Math. Anal. Appl. 444, 1640–1651 (2016)
Bès, J.: Dynamics of weighted composition operators. Compl. Anal. Oper. Theory 8, 159–176 (2014)
Bonet, J., Lindström, M., Wolf, E.: Topological structure of the set of weighted composition operators on weighted Bergman spaces of infinite order. Integr. Equ. Oper. Theory 65, 195–210 (2009)
Carswell, B., MacCluer, B., Schuster, A.: Composition operators on the Fock space. Acta Sci. Math. 69, 871–887 (2003)
Contreras, M.D., Hernández-Díaz, A.G.: Weighted composition operators between different Hardy spaces. Integr. Equ. Oper. Theory 46, 871–887 (2003)
Cowen, C.C.: The commutant of an analytic Toeplitz operator. Trans. Am. Math. Soc. 239, 1–31 (1978)
Cowen, C.C.: An analytic Toeplitz operator that commutes with a compact operator and a related class of Toeplitz operators. J. Funct. Anal. 36, 169–184 (1980)
Cowen, C.C.: A new class of operators and a description of adjoints of composition operators. J. Funct. Anal. 238, 447–462 (2006)
Cowen, C.C., MacCluer, B.D.: Composition operators on spaces of analytic functions, studies in advanced mathematics. CRC Press, Boca Raton (1995)
C̆uc̆ković, Z., Zhao, R.: Weighted composition operators on the Bergman space. J. London Math. Soc. 70(2), 499–511 (2004)
Dai, J.: Topological components of the space of composition operators on Fock spaces. Compl. Anal. Oper. Theory 9, 201–212 (2015)
Forelli, F.: The isometries of H p. Can. J. Math. 16, 721–728 (1964)
Gallardo-Gutiérrez, E.A., González, M.J., Nieminen, P.J., Saksman, E.: On the connected component of compact composition operators on the Hardy space. Adv. Math. 219, 986–1001 (2008)
Hai, P.V., Khoi, L.H.: Boundedness and compactness of weighted composition operators on Fock spaces \({\mathcal {F}}^{p}(\mathbb {C})\). Acta Math. Vietnam 41(3), 531–537 (2016)
Hai, P.V., Khoi, L.H.: Complex symmetry of weighted composition operators on the Fock space. J. Math. Anal. Appl. 433(2), 1757–1771 (2016)
Hosokawa, T., Izuchi, K., Ohno, S.: Topological structure of the space of weighted composition operators on \(H^{\infty }\). Integr. Equ. Oper. Theory 53, 509–526 (2005)
Hu, Z., Lv, X.: Toeplitz operators from one Fock space to another. Integr. Equ. Oper. Theory 70, 541–559 (2011)
Izuchi, K.J., Ohno, S.: Path connected components in weighted composition operators on \(h^{\infty }\) and \(H^{\infty }\) with the operator norm. Trans. Am. Math. Soc. 365, 3593–3612 (2013)
Izuchi, K.J., Izuchi, Y., Ohno, S.: Topological structure of the space of weighted composition operators between different Hardy spaces. Integr. Equ. Oper. Theory 80, 153–164 (2014)
Le, T.: Normal and isometric weighted composition operators on the Fock space. Bull. Lond. Math. Soc. 46, 847–856 (2014)
MacCluer, B.D., Ohno, S., Zhao, R.: Topological structure of the space of composition operators on \(H^{\infty }\). Integr. Equ. Oper. Theory 40, 481–494 (2001)
Manhas, J.S.: Compact differences of weighted composition operators on weighted Banach spaces of analytic functions. Integr. Equ. Oper. Theory 62, 419–428 (2008)
Moorhouse, J.: Compact differences of composition operators. J. Funct. Anal. 219, 70–92 (2005)
Shapiro, J.H.: Compositions operators and classical function theory. Springer, New York (1993)
Shapiro, J.H., Sundberg, C.: Isolation amongst the composition operators. Pac. J. Math. 145, 117–152 (1990)
Ueki, S.: Weighted composition operator on the Fock space. Proc. Am. Math. Soc. 135, 1405–1410 (2007)
Yousefi, B., Rezaei, H.: Hypercyclic property of weighted composition operators. Proc. Am. Math. Soc. 135, 3263–3271 (2007)
Zhu, K.: Analysis on Fock Spaces. Springer, New York (2012)
Acknowledgements
The authors would like to thank the Referee for useful remarks and comments that led to the improvement of the paper. Thanks also go to Trieu Le for useful comments on the first version of this paper.
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Supported in part by MOE’s AcRF Tier 1 grants M4011166.110 (RG24/13) and M4011724.110 (RG128/16).
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Tien, P.T., Khoi, L.H. Weighted Composition Operators Between Different Fock Spaces. Potential Anal 50, 171–195 (2019). https://doi.org/10.1007/s11118-017-9678-y
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DOI: https://doi.org/10.1007/s11118-017-9678-y