Commutants of analytic Toeplitz operators on the Bergman space

Axler, Sheldon; Čučković, Željko; Rao, N.

doi:10.1090/S0002-9939-99-05436-2

Commutants of analytic Toeplitz operators on the Bergman space
HTML articles powered by AMS MathViewer

by Sheldon Axler, Željko Čučković and N. V. Rao PDF

Proc. Amer. Math. Soc. 128 (2000), 1951-1953 Request permission

Abstract:

In this note we show that if two Toeplitz operators on a Bergman space commute and the symbol of one of them is analytic and nonconstant, then the other one is also analytic.

References

W. J. Trjitzinsky, General theory of singular integral equations with real kernels, Trans. Amer. Math. Soc. 46 (1939), 202–279. MR 92, DOI 10.1090/S0002-9947-1939-0000092-6
Sheldon Axler and Allen Shields, Algebras generated by analytic and harmonic functions, Indiana Univ. Math. J. 36 (1987), no. 3, 631–638. MR 905614, DOI 10.1512/iumj.1987.36.36034
Christopher J. Bishop, Approximating continuous functions by holomorphic and harmonic functions, Trans. Amer. Math. Soc. 311 (1989), no. 2, 781–811. MR 961619, DOI 10.1090/S0002-9947-1989-0961619-2
Arlen Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/64), 89–102. MR 160136, DOI 10.1007/978-1-4613-8208-9_{1}9
E. M. Čirka, Approximation by holomorphic functions on smooth manifolds in $\textbf {C}^{n}$, Mat. Sb. (N.S.) 78 (120) (1969), 101–123 (Russian). MR 0239121
Carl C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239 (1978), 1–31. MR 482347, DOI 10.1090/S0002-9947-1978-0482347-9
Željko Čučković, Commutants of Toeplitz operators on the Bergman space, Pacific J. Math. 162 (1994), no. 2, 277–285. MR 1251902
eljko C̆u ković and N. V. Rao, Mellin transform, monomial symbols, and commuting Toeplitz operators, J. Funct. Anal. 154 (1998), no. 1, 195–214. MR 1616532, DOI 10.1006/jfan.1997.3204
Alexander J. Izzo, Uniform algebras generated by holomorphic and pluriharmonic functions, Trans. Amer. Math. Soc. 339 (1993), no. 2, 835–847. MR 1139494, DOI 10.1090/S0002-9947-1993-1139494-6
James E. Thomson, The commutant of a class of analytic Toeplitz operators, Amer. J. Math. 99 (1977), no. 3, 522–529. MR 461196, DOI 10.2307/2373929
James Thomson, The commutant of a class of analytic Toeplitz operators. II, Indiana Univ. Math. J. 25 (1976), no. 8, 793–800. MR 417843, DOI 10.1512/iumj.1976.25.25063