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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Separation for kernels of Hankel operators


Author: Caixing Gu
Journal: Proc. Amer. Math. Soc. 129 (2001), 2353-2358
MSC (2000): Primary 47B35.
Published electronically: December 28, 2000
MathSciNet review: 1823918
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Abstract:

We prove that for two Hankel operators $H_{a_{1}}$ and $H_{a_{2}}$ on the Hardy space of the unit disk either the kernel of $H_{a_{1}}^{*}H_{a_{2}}$equals the kernel of $H_{a_{2}}$ or the kernel of $H_{a_{2}}^{*}H_{a_{1}}$equals the kernel of $H_{a_{1}}$. In fact we prove a version of the above result for products of an arbitrary finite number of Hankel operators. Some immediate corollaries are generalizations of the result of Brown and Halmos on zero products of two Hankel operators and the result of Axler, Chang and Sarason on finite rank products of two Hankel operators. Simple examples show our results are sharp.


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Additional Information

Caixing Gu
Affiliation: Department of Mathematics, California Polytechnic State University, San Luis Obispo, California 93407
Email: cgu@calpoly.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05807-X
PII: S 0002-9939(00)05807-X
Keywords: Hankel operator
Received by editor(s): May 4, 1999
Received by editor(s) in revised form: December 7, 1999
Published electronically: December 28, 2000
Additional Notes: This research was partially supported by the National Science Foundation Grant DMS-9706838 and the SFSG Grant of California Polytechnic State University.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2000 American Mathematical Society