Abstract
For a sequence of positive numbers \(\beta =\{\beta _{n}\}_{n\in \mathbb {Z}}\), the space \(L^2(\beta )\) consists of all \(f(z)=\sum _{-\infty }^\infty a_nz^n\), \(a_n\in \mathbb {C}\) for which \(\sum _{-\infty }^\infty |a_n|^2\beta _n^2<\infty \). For a bounded function \(\varphi (z)=\sum _{-\infty }^\infty a_nz^n\), the slant weighted Toeplitz operator \(A_\varphi ^{(\beta )}\) is an operator on \(L^2(\beta )\) defined as \(A_\varphi ^{(\beta )}=WM_\varphi ^{(\beta )}\), where \(M_\varphi ^{(\beta )}\) is the weighted multiplication operator on \(L^2(\beta )\) and W is an operator on \(L^2(\beta )\) such that \(Wz^{2n}=z^n\), \(Wz^{2n-1}=0\) for all \(n\in \mathbb {Z}\). In this paper we show that for a trigonometric polynomial \(\varphi (z)=\sum _{n=-p}^q a_nz^n\), \(A_\varphi ^{(\beta )}\) cannot be hyponormal unless \(\varphi \equiv 0\). We also show that, for \(k \ge 2 \) the \(k^{th}\) order slant weighted Toeplitz operator \( U_{k,\varphi }^{(\beta )}\) cannot be hyponormal unless \(\phi \equiv 0 \). Also the compression of \( U_{k,\varphi }^{(\beta )}\) to \(H^2(\beta )\), denoted by \( V_{k,\varphi }^{(\beta )}\), cannot be hyponormal unless \(\phi \equiv 0 \).
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Hazarika, M., Marik, S. Hyponormality of generalised slant weighted Toeplitz operators with polynomial symbols. Arab. J. Math. 7, 9–19 (2018). https://doi.org/10.1007/s40065-017-0183-3
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DOI: https://doi.org/10.1007/s40065-017-0183-3