## Slant Hankel operator

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

**Address.**

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

**E-mail: **sc_arora1@yahoo.co.in ruchika_masi1@yahoo.co.in

**Abstract.**

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.