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On Compactness of Products of Toeplitz Operators
Trieu Le1, Tomas Miguel Rodriguez2, Sönmez Şahutoğlu1
1Department of Mathematics & Statistics, University of Toledo, 2801 W. Bancroft, Toledo, OH 43606 USA.
Complex Analysis and Operator Theory
|November 19, 2024
Summary
We investigated the conditions under which the product of Toeplitz operators is compact. For specific symbols, compactness is achieved if and only if their product vanishes on the boundary.
Area of Science:
- Functional Analysis
- Operator Theory
- Complex Analysis
Background:
- Toeplitz operators are fundamental in various areas of mathematics and physics.
- Understanding the spectral properties of operator products is crucial for theoretical advancements.
Purpose of the Study:
- To characterize the compactness of products of Toeplitz operators.
- To investigate the relationship between symbol behavior on the boundary and operator compactness.
Main Methods:
- Analysis of Toeplitz operators with continuous symbols on the polydisc closure.
- Examination of symbol behavior on the boundary and the entire polydisc.
Main Results:
- For certain symbol classes, the product of Toeplitz operators is compact iff the product of symbols vanishes on the boundary.
- Demonstrated that vanishing on the entire polydisc does not always imply compactness for general symbols.
Conclusions:
- The study provides specific conditions for Toeplitz operator product compactness.
- Highlights the complexity of the zero product problem for Toeplitz operators, remaining an open question.
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