## Algebras on the disk and doubly commuting multiplication operators

HTML articles powered by AMS MathViewer

- by Sheldon Axler and Pamela Gorkin PDF
- Trans. Amer. Math. Soc.
**309**(1988), 711-723 Request permission

## Abstract:

We prove that a bounded analytic function $f$ on the unit disk is in the little Bloch space if and only if the uniformly closed algebra on the disk generated by ${H^\infty }$ and $\overline f$ does not contain the complex conjugate of any interpolating Blaschke product. A version of this result is then used to prove that if $f$ and $g$ are bounded analytic functions on the unit disk such that the commutator ${T_f}T_g^{\ast } - T_g^{\ast }{T_f}$ (here ${T_f}$ denotes the operator of multiplication by $f$ on the Bergman space of the disk) is compact, then $(1 - |z{|^2})\min \{ |f’ (z)|,\;|g’ (z)|\} \to 0$ as $|z| \uparrow 1$.## References

- Sheldon Axler,
*The Bergman space, the Bloch space, and commutators of multiplication operators*, Duke Math. J.**53**(1986), no. 2, 315–332. MR**850538**, DOI 10.1215/S0012-7094-86-05320-2 - Sheldon Axler, Sun-Yung A. Chang, and Donald Sarason,
*Products of Toeplitz operators*, Integral Equations Operator Theory**1**(1978), no. 3, 285–309. MR**511973**, DOI 10.1007/BF01682841 - 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 - 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
Paul Edward Budde, - Lennart Carleson,
*An interpolation problem for bounded analytic functions*, Amer. J. Math.**80**(1958), 921–930. MR**117349**, DOI 10.2307/2372840 - Lennart Carleson,
*Interpolations by bounded analytic functions and the corona problem*, Ann. of Math. (2)**76**(1962), 547–559. MR**141789**, DOI 10.2307/1970375 - Sun Yung A. Chang,
*A characterization of Douglas subalgebras*, Acta Math.**137**(1976), no. 2, 82–89. MR**428044**, DOI 10.1007/BF02392413 - John B. Conway,
*Subnormal operators*, Research Notes in Mathematics, vol. 51, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1981. MR**634507** - Theodore W. Gamelin,
*Uniform algebras*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR**0410387** - 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** - Håkan Hedenmalm,
*Thin interpolating sequences and three algebras of bounded functions*, Proc. Amer. Math. Soc.**99**(1987), no. 3, 489–495. MR**875386**, DOI 10.1090/S0002-9939-1987-0875386-8 - Kenneth Hoffman,
*Bounded analytic functions and Gleason parts*, Ann. of Math. (2)**86**(1967), 74–111. MR**215102**, DOI 10.2307/1970361
Keiji Izuchi, - Sun Yung A. Chang,
*A characterization of Douglas subalgebras*, Acta Math.**137**(1976), no. 2, 82–89. MR**428044**, DOI 10.1007/BF02392413 - G. McDonald and C. Sundberg,
*Toeplitz operators on the disc*, Indiana Univ. Math. J.**28**(1979), no. 4, 595–611. MR**542947**, DOI 10.1512/iumj.1979.28.28042 - Walter Rudin,
*Spaces of type $H^{\infty }+C$*, Ann. Inst. Fourier (Grenoble)**25**(1975), no. 1, vi, 99–125 (English, with French summary). MR**377520**
Donald Sarason, - Carl Sundberg and Thomas H. Wolff,
*Interpolating sequences for $QA_{B}$*, Trans. Amer. Math. Soc.**276**(1983), no. 2, 551–581. MR**688962**, DOI 10.1090/S0002-9947-1983-0688962-3 - A. L. Vol′berg,
*Two remarks concerning the theorem of S. Axler, S.-Y. A. Chang and D. Sarason*, J. Operator Theory**7**(1982), no. 2, 209–218. MR**658609**
Dechao Zheng,

*Support sets and Gleason parts of*$M({H^\infty })$, Ph.D. thesis, Univ. of California, Berkeley, 1982.

*Bloch functions and Hankel operators on Bergman spaces in several variables*, preprint.

*Blaschke products in*${B_0}$, Linear and Complex Analysis Problem Book, Lecture Notes in Math., vol. 1043, Springer-Verlag, Berlin, 1984. Karel Mattheus Rudolf Stroethoff,

*Characterizations of the Bloch space and related spaces*, Ph.D. thesis, Michigan State Univ., 1987.

*Hankel operators and Toeplitz operators on the Bergman space*, preprint.

## Additional Information

- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**309**(1988), 711-723 - MSC: Primary 46J15; Secondary 47B35
- DOI: https://doi.org/10.1090/S0002-9947-1988-0961609-9
- MathSciNet review: 961609