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Hankel operators; slant Hankel operators; slant Toeplitz operators
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.
[1] Arora S. C., Ruchika Batra: On Slant Hankel Operators. to appear in Bull. Calcutta Math. Soc. MR 2392281
[2] Brown A., Halmos P. R.: Algebraic properties of Toeplitz operators. J. Reine Angew. Math. 213 (1964), 89–102. MR 0160136
[3] Halmos P. R.: Hilbert Space Problem Book. Springer Verlag, New York, Heidelberg-Berlin, 1979.
[4] Ho M. C.: Properties of Slant Toeplitz operators. Indiana Univ. Math. J. 45 (1996), 843–862. MR 1422109 | Zbl 0880.47016
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