THE COMPRESSION OF A SLANT HANKEL OPERATOR TO $H^2$

Taddesse Zegeye and S. C. Arora

Abstract: A slant Hankel operator $K_{\varphi}$ with symbol $\varphi$ in $L^{\infty}(T)$ (in short $L^{\infty})$, where $T$ is the unit circle on the complex plane, is an operator whose representing matrix $M=(a_{ij})$ is given by $a_{i,j}=<\varphi,z^{-2i-j}\>$, where $<\cdot,\cdot\>$ is the usual inner product in $L^2(T)$ (in short $L^2)$. The operator $L_{\varphi}$ denotes the compression of $K_{\varphi}$ to $H^2(T)$ (in short $H^2$). We prove that an operator $L$ on $H^2$ is the compression of a slant Hankel operator to $H^2$ if and only if $U*L=LU^2$, where $U$ is the unilateral shift. Moreover, we show that a hyponormal $L_{\varphi}$ is necessarily normal and $L_{\varphi}$ can not be an isometry.

Keywords: Toeplitz poerator, slant Toeplitz operator, Hankel operator, slant Hankel operator

Classification (MSC2000): 47D99

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