The Essential Norms of Toeplitz Operators with Symbols in \(C+H^\infty \) on Weighted Hardy Spaces are Independent of the Weights

Let \(1<p<\infty \) $1<p<\infty$ , let \(H^p\) $H^p$ be the Hardy space on the unit circle, and let \(H^p(w)\) $H^p(w)$ be the Hardy space with a Muckenhoupt weight \(w\in A_p\) $w\in A_p$ on the unit circle. In 1988, Böttcher, Krupnik and Silbermann proved that the essential norm of the Toeplitz operator T(a) with \(a\in C\) $a\in C$ on the weighted Hardy space \(H^2(\varrho )\) $H^2(\varrho )$ with a power weight \(\varrho \in A_2\) $\varrho \in A_2$ is equal to \(\Vert a\Vert _{L^\infty }\) ${\Vert a\Vert }_{L^{\infty }}$ . This implies that the essential norm of T(a) on \(H^2(\varrho )\) $H^2(\varrho )$ does not depend on \(\varrho \) $\varrho$ . We extend this result and show that if \(a\in C+H^\infty \) $a\in C+H^{\infty }$ , then, for \(1<p<\infty \) $1<p<\infty$ , the essential norms of the Toeplitz operator T(a) on \(H^p\) $H^p$ and on \(H^p(w)\) $H^p(w)$ are the same for all \(w\in A_p\) $w\in A_p$ . In particular, if \(w\in A_2\) $w\in A_2$ , then the essential norm of the Toeplitz operator T(a) with \(a\in C+H^\infty \) $a\in C+H^{\infty }$ on the weighted Hardy space \(H^2(w)\) $H^2(w)$ is equal to \(\Vert a\Vert _{L^\infty }\) ${\Vert a\Vert }_{L^{\infty }}$ .