Slant Hankel operator

S. C. Arora, Ruchika Batra, M. P. Singh

Department of Mathematics, University of Delhi,  Delhi - 110 007, India



In this paper the notion of slant Hankel operator $K_\varphi$, with symbol $\varphi$ in$L^\infty$, on the space $L^2(\mathbb T)$, $\mathbb T$ being the unit circle, is introduced.The matrix of the slant Hankel operator with respect to the usual basis $\{z^i : i \in \mathbb Z \}$ of the space $L^2$ is given by $\langle\alpha_{ij}\rangle = \langle a_{-2i-j}\rangle$, where $\sum\limits_{i=-\infty}^{\infty}a_i z^i$ is the Fourier expansion of $\varphi$. Some algebraic properties such as the norm, compactness of the operator $K_\varphi$ are discussed. Along with the algebraic properties some spectral properties of such operators are discussed. Precisely, it is proved that for an invertible symbol $\varphi$, the spectrum of $K_\varphi$ contains a closed disc.

AMSclassification. 47B35.

Keywords. Hankel operators, slant Hankel operators, slant Toeplitz operators.