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Dual Toeplitz operators on the Hardy space of the unit circle are anti-unitarily equivalent to Toeplitz operators. In higher dimensions, for instance on the unit sphere, dual Toeplitz operators might behave quite differently and, therefore, seem to be a worth studying new class of Toeplitz-type operators. The purpose of this paper is to introduce and start a systematic investigation of dual Toeplitz operators on the orthogonal complement of the Hardy space of the unit sphere in Cn . In particular, we establish a corresponding spectral inclusion theorem and a Brown-Halmos type theorem. On the other hand, we characterize commuting dual Toeplitz operators as well as normal and quasinormal ones.

We characterize skew-symmetric operators on a reproducing kernel Hilbert space in terms of their Berezin symbols. The solution of some operator equations with skew-symmetric operators is studied in terms of Berezin symbols. We also studied essentially unitary operators via Berezin symbols.

The study of dual Toeplitz operators was elaborated by Stroethoff and Zheng (2002), where various corresponding algebraic and spectral properties were established. In this paper, we characterize numerical ranges of certain classes of dual Toeplitz operators. Moreover, we introduce the analog of Halmos' fifth classification problem for quasinormal dual Toeplitz operators. In particular, we show that there are no quasinormal dual Toeplitz operators with bounded analytic or coanalytic symbols which are not normal.

In this paper, we construct a symbol calculus yielding short exact sequences for the dual Toeplitz algebra generated by all bounded dual Toeplitz operators on the Hardy space associated with the polydisk Dn in the unitary space Cn, that have been introduced and well studied in our earlier paper (Benaissa and Guediri in Taiwan J Math 19: 31–49, 2015), as well as for the C*-subalgebra generated by dual Toeplitz operators with symbols continuous on the associated hypertorus Tn.

In this paper, we introduce dual Toeplitz operators on the orthogonal complement of the Hardy space of the polydisk and establish their main algebraic properties using an auxiliary transformation of operators. As a byproduct, we exploit this mysterious transformation in the investigation of boundedness and compactness of Hankel products and mixed Toeplitz-Hankel products on the Hardy space of the polydisk. 2010Mathematics Subject Classification: 47B35. Key words and phrases: Dual Toeplitz operator, Hardy space of the polydisk, Commuting, Brown-Halmos, Hankel products, Mixed Toeplitz-Hankel products.

The study of dual Toeplitz operators was elaborated by Stroethoff and Zheng (2002), where various corresponding algebraic and spectral properties were established. In this paper, we characterize numerical ranges of certain classes of dual Toeplitz operators. Moreover, we introduce the analog of Halmos' fifth classification problem for quasinormal dual Toeplitz operators. In particular, we show that there are no quasinormal dual Toeplitz operators with bounded analytic or coanalytic symbols which are not normal.

For a bounded linear operator \\(A\\) on a reproducing kernel Hilbert space \\(\\mathcal{H}(\\Omega)\\), with normalized reproducing kernel \\(\\widehat{k}_{\\lambda} = \\frac{k_{\\lambda}}{\\lVert k_{\\lambda}\\lVert}\\), the Berezin symbol, Berezin number and Berezin norm are defined respectively by \\(\\widetilde{A}(\\lambda) = \\langle A\\widehat{k}_{\\lambda},\\widehat{k}_{\\lambda}\\rangle\\), \\(ber(A) = \\sup_{\\lambda\\in\\Omega}\\left|\\widetilde{A}(\\lambda)\\right|\\) and \\(\\left\\|A\\right\\|_{ber} = \\sup_{\\lambda\\in\\Omega}\\left\\|A\\widehat{k}_{\\lambda}\\right\\|\\). A straightforward comparison between these characteristics yields the inequalities \\(ber(A)\\leq\\left\\|A\\right\\|_{ber}\\leq\\lVert A\\lVert\\). In this paper, we prove further inequalities relating them, and give special care to the corresponding reverse inequalities. In particular, we refine the first one of the above inequalities, namely we prove that \\ \\(ber(A)\\leq\\left( \\left\\|A\\right\\|_{ber}^{2}-\\inf_{\\lambda\\in\\Omega}\\left\\lVert (A-\\widetilde{A}(\\lambda))\\widehat{k}_{\\lambda}\\right\\lVert^{2}\\right) ^{\\frac{1}{2}}\\).