Compact Composition Operators on BMOA

Bourdon, P.; Cima, J.; Matheson, A.

doi:10.1090/S0002-9947-99-02387-9

Compact Composition Operators on BMOA
HTML articles powered by AMS MathViewer

by P. S. Bourdon, J. A. Cima and A. L. Matheson PDF

Trans. Amer. Math. Soc. 351 (1999), 2183-2196 Request permission

Abstract:

We characterize the compact composition operators on BMOA, the space consisting of those holomorphic functions on the open unit disk $U$ that are Poisson integrals of functions on $\partial U$, that have bounded mean oscillation. We then use our characterization to show that compactness of a composition operator on BMOA implies its compactness on the Hardy spaces (a simple example shows the converse does not hold). We also explore how compactness of the composition operator $C_\phi : \operatorname {BMOA}\rightarrow \operatorname {BMOA}$ relates to the shape of $\phi (U)$ near $\partial U$, introducing the notion of mean order of contact. Finally, we discuss the relationships among compactness conditions for composition operators on BMOA, VMOA, and the big and little Bloch spaces.

References

J. M. Anderson, J. Clunie, and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12–37. MR 361090
Joseph A. Cima and Alec L. Matheson, Essential norms of composition operators and Aleksandrov measures, Pacific J. Math. 179 (1997), no. 1, 59–64. MR 1452525, DOI 10.2140/pjm.1997.179.59
Carl C. Cowen and Barbara D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397026
N. Dunford and J. Schwartz, Linear Operators, John Wiley and Sons, New York, 1988.
P. L. Duren, B. W. Romberg, and A. L. Shields, Linear functionals on $H^{p}$ spaces with $0<p<1$, J. Reine Angew. Math. 238 (1969), 32–60. MR 259579
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
Kevin Madigan and Alec Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (1995), no. 7, 2679–2687. MR 1273508, DOI 10.1090/S0002-9947-1995-1273508-X
Donald Sarason, Function theory on the unit circle, Virginia Polytechnic Institute and State University, Department of Mathematics, Blacksburg, Va., 1978. Notes for lectures given at a Conference at Virginia Polytechnic Institute and State University, Blacksburg, Va., June 19–23, 1978. MR 521811
Joel H. Shapiro, The essential norm of a composition operator, Ann. of Math. (2) 125 (1987), no. 2, 375–404. MR 881273, DOI 10.2307/1971314
Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1237406, DOI 10.1007/978-1-4612-0887-7
J. H. Shapiro and P. D. Taylor, Compact, nuclear, and Hilbert-Schmidt composition operators on $H^{2}$, Indiana Univ. Math. J. 23 (1973/74), 471–496. MR 326472, DOI 10.1512/iumj.1973.23.23041
W. Smith, Inner functions in the hyperbolic little Bloch class, Michigan Math. J. 45 (1988), 103–104.
Kenneth Stephenson, Weak subordination and stable classes of meromorphic functions, Trans. Amer. Math. Soc. 262 (1980), no. 2, 565–577. MR 586736, DOI 10.1090/S0002-9947-1980-0586736-2
M. Tjani, Compact composition operators on some Möbius invariant Banach spaces, Thesis, Michigan State University, 1996.
P. Wojtaszczyk, Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, vol. 25, Cambridge University Press, Cambridge, 1991. MR 1144277, DOI 10.1017/CBO9780511608735