Slant Hankel operators

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

Archivum mathematicum, Tome 42 (2006) no. 2, pp. 125-133.

Résumé

In this paper the notion of slant Hankel operator $K_\varphi$, with symbol $\varphi$ in $L^\infty$, on the space $L^2({\Bbb T})$, ${\Bbb T}$ being the unit circle, is introduced. The matrix of the slant Hankel operator with respect to the usual basis $\{z^i : i \in {\Bbb Z} \}$ 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.

Détail

Classification : 47A10, 47B35
Keywords: Hankel operators; slant Hankel operators; slant Toeplitz operators