Generalized composition operators on Zygmund spaces and Bloch type spaces. (English) Zbl 1135.47021

Li, Songxiao; Stević, Stevo

J. Math. Anal. Appl. 338, No. 2, 1282-1295 (2008).

Let \(D\) denote the unit disc in the complex plane, let \(H(D)\) denote the set of all functions holomorphic on \(D\), and let \(C(\overline{D})\) denote the set of all functions continuous on the closure of \(D\). A Bloch type space (or \(\alpha\)-Bloch space) is a space of the form \[ B^{\alpha} = \{f \in H(D): \sup_{z \in D} (1 - | z| ^{2})^{\alpha} | f'(z)| < \infty \}, \] where the space \(B^{\alpha}\) is given the norm \[ \| f\| _{B^{\alpha}} = | f(0)| + \sup_{z \in D} (1 - | z| ^{2})^{\alpha} | f'(z)|. \] The Zygmund space \(Z\) is the space \[ Z = \left\{f \in H(D) \cap C(\overline{D}): \sup_{\theta \in [0, 2\pi], h > 0} \frac {| f(e^{i \theta + h}) + f(e^{i \theta - h}) - 2 f(e^{i \theta})| } {h} < \infty \right\}, \] with the norm given by \[ \| f\| _{Z} = | f(0)| + | f'(0)| + \sup_{z \in D} (1 - | z| ^{2}) | f''(z)|. \] Throughout, \(\varphi\) denotes a non-constant analytic self-map of \(D\). A basic composition operator is given by \(C_{\varphi}f = f \circ \varphi\) for \(f \in H(D)\). Let \(g \in H(D)\) and define the linear operator \[ (C_{\varphi}^{g}f)(z) = \int_{0}^{z} f'(\varphi(\zeta))g(\zeta)\,d\zeta. \] The authors give criteria under which the general composition operator \(C_{\varphi}^{g}:Z \to B^{\alpha}\) is a bounded operator, and also when it is a compact operator. Also considered are the cases when \(C_{\varphi}^{g}: Z \to Z\) and when \(C_{\varphi}^{g}: B^{\alpha} \to Z\) is a bounded operator, and when it is a compact operator. Letting \[ B_{0}^{\alpha} = \left\{f \in B^{\alpha}: \lim_{| z| \to 1} \;(1 - | z| ^{2})^{\alpha}| f'(z) = 0 \right\}, \] and letting \[ Z_{0} = \left\{f \in Z: \lim_{| z| \to 1} (1 - | z| ^{2})| f''(z)| = 0 \right\}, \] corresponding results are obtained using \(B_{0}^{\alpha}\) in place of \(B^{\alpha}\) and using \(Z_{0}\) in place of \(Z\). Two typical results are as follows. Theorem. If \(0 < \alpha < \infty\), if \(g \in H(D)\) and \(\varphi\) is an analytic self-map of \(D\), then \(C_{\varphi}^{g}: Z \to B^{\alpha}\) is bounded if and only if \[ \sup_{z\in D}\,(1-| z|^{2})^{\alpha}| g(z)| \log\frac{1}{1-|\varphi(z)|^{2}}<\infty. \] In addition, this operator is compact if and only it is bounded and \[ \lim_{| \varphi(z)| \to 1} \;(1 - | z| ^{2})^{\alpha} | g(z)| \log \frac {1} {1 - | \varphi(z)| ^{2}} = 0. \] Theorem. If \(0 < \alpha < \infty\), if \(g \in H(D)\), and if \(\varphi\) is an analytic self-map of \(D\), then the following statements are equivalent: (i) \(C_{\varphi}^{g}: B^{\alpha} \to Z\) is compact; (ii) \(C_{\varphi}^{g}: B_{0}^{\alpha} \to Z\) is compact; (iii) \(C_{\varphi}^{g}: B^{\alpha} \to Z\) is bounded and both \[ \lim_{| \varphi(z)| \to 1} \frac {(1 - | z| ^{2}) | \varphi'(z)|\,| g(z)|} {(1 - | z|^{2})^{\alpha + 1}} = 0\quad\text{and}\quad\lim_{| \varphi(z)| \to 1} \frac {(1-| z|^{2})| g'(z)|}{(1-| \varphi(z)| ^{2})^{\alpha}} = 0. \]

Reviewer: Peter Lappan (East Lansing)

MSC:

47B33	Linear composition operators
30D45	Normal functions of one complex variable, normal families
47B38	Linear operators on function spaces (general)
46E15	Banach spaces of continuous, differentiable or analytic functions

Keywords:

Zygmund space; Bloch type space; generalised composition operator

References:

[1]	Boe, B.; Nikolau, A., Interpolation by functions in the Bloch space, J. anal. math., 94, 171-194, (2004) · Zbl 1094.30042
[2]	Choe, B.; Koo, H.; Smith, W., Composition operators on small spaces, Integral equations operator theory, 56, 357-380, (2006) · Zbl 1114.47028
[3]	Cowen, C.; MacCluer, B., Composition operators on spaces of analytic functions, Stud. adv. math., (1995), CRC Press Boca Raton · Zbl 0873.47017
[4]	Danford, N.; Schwartz, J.T., Linear operators I, (1958), Interscience Publishers, John Willey and Sons New York
[5]	Duren, P., Theory of \(H^p\) spaces, (1973), Academic Press New York · Zbl 0215.20203
[6]	Hu, Z., Extended Cesàro operators on the Bloch space in the unit ball of \(\mathbb{C}^n\), Acta math. sci. ser. B engl. ed., 23, 4, 561-566, (2003) · Zbl 1044.47023
[7]	Hornor, W.; Jamison, J.E., Isometries of some Banach spaces of analytic functions, Integral equations operator theory, 41, 401-425, (2001) · Zbl 0995.46012
[8]	Li, S.; Stević, S., Volterra-type operators on Zygmund spaces, J. ineq. appl., 2007, (2007), Article ID 32124, 10 pp., doi:10.1155/2007/32124
[9]	Lou, Z., Composition operators on Bloch type spaces, Analysis (Munich), 23, 1, 81-95, (2003) · Zbl 1058.47024
[10]	Madigan, K.; Matheson, A., Compact composition operators on the Bloch space, Trans. amer. math. soc., 347, 7, 2679-2687, (1995) · Zbl 0826.47023
[11]	Ohno, S.; Stroethoff, K.; Zhao, R., Weithted composition operators between Bloch type spaces, Rocky mountain J. math., 33, 191-215, (2003) · Zbl 1042.47018
[12]	Shapiro, J., Composition operators and classical function theory, (1993), Springer-Verlag New York · Zbl 0791.30033
[13]	Stević, S., On an integral operator on the unit ball in \(\mathbb{C}^n\), J. inequal. appl., 2005, 1, 81-88, (2005) · Zbl 1074.47013
[14]	Xiao, J., Riemann – stieltjes operators on weighted Bloch and Bergman spaces of the unit ball, J. London math. soc., 70, 2, 199-214, (2004) · Zbl 1064.47034
[15]	Xiao, J., Composition operators associated with Bloch-type spaces, Complex var. theory appl., 46, 109-121, (2001) · Zbl 1044.47020
[16]	Zhu, K., Bloch type spaces of analytic functions, Rocky mountain J. math., 23, 3, 1143-1177, (1993) · Zbl 0787.30019
[17]	Zhu, K., Operator theory in function spaces, (1990), Marcel Dekker New York