Abstract
In this paper, we generalize the concept of asymptotic Hankel operators on \(H^2(\mathbb{D})\) to the Hardy space \(H^2(\mathbb{D}^n)\) (over polydisk) in terms of asymptotic Hankel and partial asymptotic Hankel operators and investigate some properties in case of its weak and strong convergence. Meanwhile, we introduce ith-partial Hankel operators on \(H^2(\mathbb{D}^n)\) and obtain a characterization of its compactness for n > 1. Our main results include the containment of Toeplitz algebra in the collection of all strong partial asymptotic Hankel operators on \(H^2(\mathbb{D}^n)\). It is also shown that a Toeplitz operator with symbol ϕ is asymptotic Hankel if and only if ϕ is holomorphic function in \(L^\infty(\mathbb{T}^n)\).
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Authors are grateful to the referees for their valuable suggestions and comments which help us in improving the manuscript.
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Support of UGC Research Grant [Ref. No.: 21/12/2014(ii) EU-V, Sr. No. 2121440601] to second author for carrying out the research work is gratefully acknowledged
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Gupta, A., Gupta, B. Asymptotic and Partial Asymptotic Hankel Operators on \(H^2(\mathbb{D}^n)\)). Acta. Math. Sin.-English Ser. 35, 1729–1740 (2019). https://doi.org/10.1007/s10114-019-8331-7
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DOI: https://doi.org/10.1007/s10114-019-8331-7
Keywords
- Hankel operator
- Asymptotic Hankel operator
- partial asymptotic Hankel operator
- i th-partial Hankel operator
- i th-partial big Hankel operator