×

On compactness of the difference of composition operators. (English) Zbl 1072.47021

Let \(\phi\) and \(\psi\) be analytic self-maps of the unit disc, and denote by \(C_\phi\) and \(C_\psi\) the induced composition operators. The authors study the compactness of the difference operator \(T=C_\phi-C_\psi\) on the Hardy spaces \(H^p\), the Lebesgue spaces \(L^p\), and the measure space \(M\) of the unit circle. The authors prove several results described below. 1. (Compactness on \(H^p\)) It is shown that the compactness of \(T\) on \(H^p\) is independent of \(p\in[1,\infty)\) and equivalent to the weak compactness on \(H^1\). However, the authors do not provide any characterization in terms of symbols. 2. (Compactness on \(L^1\) and \(M\)) It is shown that \(T\) is either compact or weakly compact on \(L^1\) if and only if \(T\) is either compact or weakly compact on \(M\). Also, a characterization in terms of the Aleksandrov measures of symbols is provided. 3. (Different nature of \(H^1\) and \(L^1\)) It is well-known that the compactness of a single composition operator on \(H^1\) is equivalent to the compactness on \(L^1\) or \(M\). It is shown that this phenomenon does not extend to difference operators. The authors construct an explicit counterexample \(T\) which is compact on \(H^1\), but not on \(L^1\). The construction is substantially complicated and relies on somewhat delicate estimates involving harmonic measures. 4. The authors construct another example of a non-compact operator \(T\) on \(H^1\) for which the singular parts of the Aleksandrov measures of symbols coincide everywhere on the unit circle. This shows that there is no redundant condition in the Aleksandrov measure characterization of compactness on \(L^1\) mentioned above. In addition, this example disproves a conjecture posed by J. E. Shapiro [J. Oper. Theory 40, 133–146 (1998; Zbl 0997.47023)]. 5. (Compactness on \(H^\infty\) and \(L^\infty\)) \(T\) is either compact or weakly compact on \(H^\infty\) if and only if \(T\) is either compact or weakly compact on \(L^\infty\). This extends an earlier characterization of the compactness of \(T\) on \(H^\infty\) given by B. McCluer, S. Ohno and R. Zhao [Integral Equations Oper. Theory 40, 481–494 (2001; Zbl 1062.47511)].

MSC:

47B33 Linear composition operators
30D55 \(H^p\)-classes (MSC2000)
47B38 Linear operators on function spaces (general)
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] Aleksandrov, A.B., The multiplicity of boundary values of inner functions, Izv. akad. nauk armyanskoi SSR ser. mat, 22, 490-503, (1987), (in Russian) · Zbl 0648.30002
[2] Bennet, C.; Sharpley, R., Interpolation of operators, Pure appl. math, vol. 129, (1988), Academic Press Boston, MA
[3] Bourdon, P., Components of linear-fractional composition operators, J. math. anal. appl, 279, 228-245, (2003) · Zbl 1043.47021
[4] Cima, J.A.; Matheson, A.L., Essential norms of composition operators and Aleksandrov measures, Pacific J. math, 179, 59-63, (1997) · Zbl 0871.47027
[5] Cima, J.A.; Matheson, A.L., Cauchy transforms and composition operators, Illinois J. math, 42, 58-69, (1998) · Zbl 0914.30023
[6] Cowen, C.; MacCluer, B.D., Composition operators on spaces of analytic functions, (1995), CRC Press Boca Raton, FL · Zbl 0873.47017
[7] Cwikel, M., Real and complex interpolation and extrapolation of compact operators, Duke math. J, 65, 333-343, (1992) · Zbl 0787.46062
[8] Diestel, J., A survey of results related to the dunford – pettis property, Contemp. math, 2, 15-60, (1980)
[9] Dunford, N.; Schwartz, J.T., Linear operators, part I, (1958), Interscience New York
[10] Fefferman, C.; Rivière, N.M.; Sagher, Y., Interpolation between Hp spaces: the real method, Trans. amer. math. soc, 191, 75-81, (1974) · Zbl 0285.41006
[11] Garnett, J.B., Bounded analytic functions, (1981), Academic Press New York · Zbl 0469.30024
[12] Goebeler, T.E., Composition operators acting between Hardy spaces, Integral equations operator theory, 41, 389-395, (2001) · Zbl 0997.47021
[13] T. Hosokawa, K. Izuchi, S. Ohno, Topological structure of the space of weighted composition operators on H∞, manuscript · Zbl 1098.47025
[14] Katznelson, Y., An introduction to harmonic analysis, (1968), Wiley New York, reprinted by Dover, New York, 1976 · Zbl 0169.17902
[15] MacCluer, B.D., Components in the space of composition operators, Integral equations operator theory, 12, 725-738, (1989) · Zbl 0685.47027
[16] MacCluer, B.; Ohno, S.; Zhao, R., Topological structure of the space of composition operators on H∞, Integral equations operator theory, 40, 481-494, (2001) · Zbl 1062.47511
[17] Moorhouse, J.; Toews, C., Differences of composition operators, Trends in Banach spaces and operator theory, Memphis, TN, 2001, Contemp. math, 321, 207-213, (2003) · Zbl 1052.47018
[18] Nieminen, P.J.; Saksman, E., Boundary correspondence of Nevanlinna counting functions for self-maps of the unit disc, Trans. amer. math. soc, 356, 3167-3187, (2004) · Zbl 1210.30012
[19] Rudin, W., Real and complex analysis, (1987), McGraw-Hill New York · Zbl 0925.00005
[20] Rudin, W., Functional analysis, (1991), McGraw-Hill New York · Zbl 0867.46001
[21] Sarason, D., Composition operators as integral operators, Analysis and partial differential equations, (1990), Dekker New York · Zbl 0712.47026
[22] Sarason, D., Weak compactness of holomorphic composition operators on H1, (), 75-79 · Zbl 0776.47016
[23] Shapiro, J.E., Aleksandrov measures used in essential norm inequalities for composition operators, J. operator theory, 40, 133-146, (1998) · Zbl 0997.47023
[24] Shapiro, J.H., The essential norm of a composition operator, Ann. of math, 125, 375-404, (1987) · Zbl 0642.47027
[25] Shapiro, J.H., Composition operators and classical function theory, (1993), Springer-Verlag New York · Zbl 0791.30033
[26] Shapiro, J.H.; Sundberg, C., Compact composition operators on L1, Proc. amer. math. soc, 108, 443-449, (1990) · Zbl 0704.47018
[27] Shapiro, J.H.; Sundberg, C., Isolation amongst the composition operators, Pacific J. math, 145, 117-152, (1990) · Zbl 0732.30027
[28] Shapiro, J.H.; Taylor, P.D., Compact, nuclear, and hilbert – schmidt composition operators on H2, Indiana univ. math. J, 23, 471-496, (1973) · Zbl 0276.47037
[29] Wojtaszczyk, P., Banach spaces for analysts, (1991), Cambridge Univ. Press Cambridge, UK · Zbl 0724.46012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.