## $C^\ast$-algebras generated by a subnormal operator

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- by Kit C. Chan and Z̆eljko C̆uc̆ković PDF
- Trans. Amer. Math. Soc.
**351**(1999), 1445-1460 Request permission

## Abstract:

Using the functional calculus for a normal operator, we provide a result for generalized Toeplitz operators, analogous to the theorem of Axler and Shields on harmonic extensions of the disc algebra. Besides that result, we prove that if $T$ is an injective subnormal weighted shift, then any two nontrivial subspaces invariant under $T$ cannot be orthogonal to each other. Then we show that the $C^*$-algebra generated by $T$ and the identity operator contains all the compact operators as its commutator ideal, and we give a characterization of that $C^*$-algebra in terms of generalized Toeplitz operators. Motivated by these results, we further obtain their several-variable analogues, which generalize and unify Coburn’s theorems for the Hardy space and the Bergman space of the unit ball.## References

- 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 - Hari Bercovici, Ciprian Foias, and Carl Pearcy,
*Dual algebras with applications to invariant subspaces and dilation theory*, CBMS Regional Conference Series in Mathematics, vol. 56, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1985. MR**787041**, DOI 10.1090/cbms/056 - John Bunce,
*Characters on singly generated $C^{\ast }$-algebras*, Proc. Amer. Math. Soc.**25**(1970), 297–303. MR**259622**, DOI 10.1090/S0002-9939-1970-0259622-4 - John Bunce,
*The joint spectrum of commuting nonnormal operators*, Proc. Amer. Math. Soc.**29**(1971), 499–505. MR**283602**, DOI 10.1090/S0002-9939-1971-0283602-7 - L. A. Coburn,
*Singular integral operators and Toeplitz operators on odd spheres*, Indiana Univ. Math. J.**23**(1973/74), 433–439. MR**322595**, DOI 10.1512/iumj.1973.23.23036 - John B. Conway,
*Subnormal operators*, Research Notes in Mathematics, vol. 51, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1981. MR**634507** - A. T. Dash,
*Joint spectra*, Studia Math.**45**(1973), 225–237. MR**336381**, DOI 10.4064/sm-45-3-225-237 - Peter L. Duren,
*Theory of $H^{p}$ spaces*, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR**0268655** - Jacques Dixmier,
*$C^*$-algebras*, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR**0458185** - Richard Frankfurt,
*Subnormal weighted shifts and related function spaces*, J. Math. Anal. Appl.**52**(1975), no. 3, 471–489. MR**482334**, DOI 10.1016/0022-247X(75)90074-8 - Richard Frankfurt,
*Subnormal weighted shifts and related function spaces. II*, J. Math. Anal. Appl.**55**(1976), no. 1, 2–17. MR**482335**, DOI 10.1016/0022-247X(76)90273-0 - Richard Frankfurt,
*Function spaces associated with radially symmetric measures*, J. Math. Anal. Appl.**60**(1977), no. 2, 502–541. MR**507896**, DOI 10.1016/0022-247X(77)90039-7 - T. W. Gamelin,
*Uniform Algebras,*Chelsea, New York, 1984. - P. R. Halmos,
*Introduction to Hilbert Space and the Theory of Spectral Multiplicity,*Second Edition, Chelsea, New York, 1957. - Paul Richard Halmos,
*A Hilbert space problem book*, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 17, Springer-Verlag, New York-Berlin, 1982. MR**675952**, DOI 10.1007/978-1-4684-9330-6 - Nicholas P. Jewell,
*Multiplication by the coordinate functions on the Hardy space of the unit sphere in $\textbf {C}^{n}$*, Duke Math. J.**44**(1977), no. 4, 839–851. MR**463966** - Gerard E. Keough,
*Subnormal operators, Toeplitz operators and spectral inclusion*, Trans. Amer. Math. Soc.**263**(1981), no. 1, 125–135. MR**590415**, DOI 10.1090/S0002-9947-1981-0590415-6 - Steven G. Krantz,
*Function theory of several complex variables*, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR**635928** - Arthur Lubin,
*Weighted shifts and products of subnormal operators*, Indiana Univ. Math. J.**26**(1977), no. 5, 839–845. MR**448139**, DOI 10.1512/iumj.1977.26.26067 - Heydar Radjavi and Peter Rosenthal,
*Invariant subspaces*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77, Springer-Verlag, New York-Heidelberg, 1973. MR**0367682**, DOI 10.1007/978-3-642-65574-6 - Walter Rudin,
*Real and complex analysis*, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR**924157** - Walter Rudin,
*Function theory in the unit ball of $\textbf {C}^{n}$*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR**601594**, DOI 10.1007/978-1-4613-8098-6 - Donald Sarason,
*The $H^{p}$ spaces of an annulus*, Mem. Amer. Math. Soc.**56**(1965), 78. MR**188824** - Allen L. Shields,
*Weighted shift operators and analytic function theory*, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. MR**0361899**

## Additional Information

**Kit C. Chan**- Affiliation: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403-0221
- Email: kchan@bgnet.bgsu.edu
**Z̆eljko C̆uc̆ković**- Affiliation: Department of Mathematics, University of Toledo, Toledo, Ohio 43606-3390
- MR Author ID: 294593
- Email: zcuckovi@math.utoledo.edu
- Received by editor(s): December 4, 1995
- Received by editor(s) in revised form: March 4, 1998
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**351**(1999), 1445-1460 - MSC (1991): Primary 47B20, 32A37, 46L05; Secondary 46E20, 47A13, 47B37
- DOI: https://doi.org/10.1090/S0002-9947-99-02389-2
- MathSciNet review: 1624093