# Hyponormality of generalised slant weighted Toeplitz operators with polynomial symbols

Arabian Journal of Mathematics, Sep 2017

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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Munmun Hazarika, Sougata Marik. Hyponormality of generalised slant weighted Toeplitz operators with polynomial symbols, Arabian Journal of Mathematics, 2017, 9-19, DOI: 10.1007/s40065-017-0183-3