Continuity of the norm of a~composition operator between weighted Banach spaces of holomorphic functions

Elke Wolf

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2008), pp. 99-107.

Résumé

We consider composition operators $C_{\varphi }$ between given weighted Banach spaces of analytic functions defined on the open unit disk and explore the continuity of the map, which given an analytic self-map of the disk, takes as its value the associated composition operator resp. the norm of this operator.

Bibliographie

[1] Bierstedt K.D., Summers W.H., “Biduals of weighted Banach spaces of analytic functions”, J. Austral. Math. Soc., Series A, 54 (1993), 70–79 | DOI | MR | Zbl

[2] Bierstedt K.D., Bonet J., Galbis A., “Weighted spaces of holomorphic functions on bounded domains”, Michigan Math. J., 40 (1993), 271–297 | DOI | MR | Zbl

[3] Bierstedt K.D., Bonet J., Taskinen J., “Associated weights and spaces of holomorphic functions”, Studia Math., 127 (1998), 137–168 | MR | Zbl

[4] Bonet J., Domański P., Lindström M., “Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions”, Canad. Math. Bull., 42:2 (1999), 139–148 | MR | Zbl

[5] Bonet J., Domański P., Lindström M., Taskinen J., “Composition operators between weighted Banach spaces of analytic functions”, J. Austral. Math. Soc., Serie A, 64 (1998), 101–118 | DOI | MR | Zbl

[6] Bonet J., Friz M., Jordá E., “Composition operators between weighted inductive limits of spaces of holomorphic functions”, Publ. Math., 67:3-4 (2005), 333–348 | MR | Zbl

[7] Bonet J., Lindström M., Wolf E., “Differences of composition operators between weighted Banach spaces of holomorphic functions”, J. Austral. Math. Soc., 84:1 (2008), 9–20 | DOI | MR | Zbl

[8] Cowen C., MacCluer B., Composition Operators on Spaces of Analytic Functions, CRC Press, Baca Raton, 1995 | MR | Zbl

[9] Domański P., Lindström M., “Sets of interpolation and sampling for weighted Banach spaces of holomorphic functions”, Ann. Pol. Math., 79:3 (2002), 233–264 | DOI | MR | Zbl

[10] Galbis A., “Weighted Banach spaces of entire functions”, Arch. Math., 62 (1994), 58–64 | DOI | MR | Zbl

[11] Garnett J., Bounded Analytic Functions, Academic Press, New York, 1981 | MR | Zbl

[12] Hoffman K., Banach spaces of analytic functions, Prentice Hall, Englewood Cliffs, 1967 | MR

[13] Jarchow H., “Some functional analytic properties of composition operators”, Quaestiones Math., 18 (1995), 229–256 | MR | Zbl

[14] Kaballo W., “Lifting-Probleme für $H^{\infty}$-Funktionen”, Arch. Math., 34 (1980), 540–549 | DOI | MR | Zbl

[15] Lindström M., Wolf E., “Essential norm of differences of weighted composition operators”, Monatshefte Math., 153:2 (2008), 133–143 | DOI | MR | Zbl

[16] Lusky W., “On the structure of $Hv_0(D)$ and $hv_0(D)$”, Math. Nachr., 159 (1992), 279–289 | DOI | MR | Zbl

[17] Lusky W., “On weighted spaces of harmonic and holomorphic functions”, J. London Math. Soc., 51 (1995), 309–320 | MR | Zbl

[18] MacCluer B., Ohno S., Zhao R., “Topological structure of the space of composition operators on $H^{\infty}$”, Integral Equations Operator Theory, 40:4 (2001), 481–494 | DOI | MR | Zbl

[19] Rubel L.A., Shields A.L., “The second duals of certain spaces of analytic functions”, J. Austral. Math. Soc., 11 (1970), 276–280 | DOI | MR | Zbl

[20] Pokorny D., Shapiro J., “Continuity of the norm of a composition operator”, Integral equations Oper. Theory, 45:3 (2003), 351–358 | DOI | MR | Zbl

[21] Shapiro J.H., Composition Operators and Classical Function Theory, Springer, 1993 | MR

[22] Shields A.L., Williams D.L., “Bounded projections, duality, and multipliers in spaces of harmonic functions”, J. Reine Angew. Math., 299/300 (1978), 256–279 | MR | Zbl

[23] Shields A.L., Williams D.L., “Bounded projections and the growth of harmonic conjugates in the disc”, Michigan Math. J., 29 (1982), 3–25 | DOI | MR | Zbl

[24] Wolf E., “Weighted Fréchet spaces of holomorphic functions”, Studia Math., 174:3 (2006), 255–275 | DOI | MR | Zbl