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Weighted composition operators on Hardy spaces. (English) Zbl 1026.47016
This paper studies operators of the form $f\mapsto (f\circ\varphi)\psi$ acting on Hardy spaces $H^p$ of the unit disk $D$, where $\psi$ is analytic in $D$ and $\varphi$ is an analytic self-map of $D$. Problems studied include the boundedness, compactness, weak compactness, and complete continuity of such operators. In particular, it is shown that such an operator is compact on $H^1$ if and only if it is weakly compact on $H^1$.
Reviewer: K.Zhu (Albany)

47B33Composition operators
Full Text: DOI
[1] Cima, J. A.; Matheson, A.: Completely continuous composition operators. Trans. amer. Math. soc. 344, 849-856 (1994) · Zbl 0813.47032
[2] Contreras, M. D.; Dı\acute{}az-Madrigal, S.: Compact-type operators defined on h\infty. Contemp. math. 232, 111-118 (1999) · Zbl 0936.46010
[3] Cowen, C.; Maccluer, B.: Composition operators on spaces of analytic functions. (1995) · Zbl 0873.47017
[4] Dı\acute{}az, S.: Weak compactness in L1({$\mu$},X). Proc. amer. Math. soc. 124, 2685-2693 (1996) · Zbl 0865.46024
[5] Diestel, J.: Sequences and series in Banach spaces. Graduate texts in mathematics 92 (1984) · Zbl 0542.46007
[6] Dunford, N.; Schwartz, J. T.: Linear operators, part I. (1958) · Zbl 0084.10402
[7] Forelli, F.: The isometries of hp. Canad. J. Math. 16, 721-728 (1964) · Zbl 0132.09403
[8] Halmos, P. R.: Measure theory. Graduate texts in mathematics 18 (1974)
[9] Hoffman, K.: Banach spaces of analytic functions. (1988) · Zbl 0734.46033
[10] Maccluer, B.: Compact composition operators on $Hp(BN)$. Michigan math. J. 32, 237-248 (1985) · Zbl 0585.47022
[11] Maccluer, B.; Shapiro, J.: Angular derivatives and compact composition operators on the Hardy and Bergman spaces. Canad. J. Math. 38, 878-906 (1986) · Zbl 0608.30050
[12] Mirzakarimi, G.; Seddighi, K.: Weighted composition operators on Bergman and Dirichlet spaces. Georgian math. J. 4, 373-383 (1997) · Zbl 0891.47018
[13] Sarason, D.: Weak compactness of holomorphic composition operators on H1. (1992) · Zbl 0776.47016
[14] Wojtaszczyk, P.: Banach spaces for analysts. Cambridge studies in advanced mathematics 25 (1991) · Zbl 0724.46012