Spectra of slant Toeplitz operators with continuous symbols
Let \(\varphi\) be a bounded function on the unit circle \(T\), and \(c_n\), \(n\in\mathbb{Z}\), its \(n\)th Fourier coefficient. The slant Toeplitz operator \(A_\varphi\) is an operator on \(L^2(T)\) whose representing matrix is: \[ \left[\begin{matrix} \cdots &&&&&&\cdots\\ \cdots & c_0 & c_{-1} & c_{-2} &&& \cdots\\ \cdots && c_1 & c_0 & c_{-1} && \cdots\\ \cdots &&& c_2 & c_1 & c_0 &\cdots\\ \cdots &&&&&& \cdots\end{matrix}\right]. \] Then \(A_\varphi= A_1 M_\varphi\), where \(A_1\) is the adjoint of the composition operator induced by the map \(t(\exp(i\phi))= \exp(2i\phi)\) on \(T\), and \(M_\varphi\) is the multiplication operator. In this paper, the author proves that: If \(\varphi\) is a nonvanishing continuous function on \(T\), then both the spectrum and the essential spectrum of \(A_\varphi\) are the closed disk \(\{z\in\mathbb{C}; | z|\leq r(A_\varphi)\}\). If \(| z|< r(A_\varphi)\), then \(\dim\ker(A_\varphi- z)= \infty\), and \(\dim\ker(A^*_\varphi-\overline z)= 0\). When \(\varphi\) is a nonvanishing trigonometric polynomial on \(T\), this result is obtained by Latushkin in 1982, by a different method.
scientific article; zbMATH DE number 1046974