The essential norms of Toeplitz operators with symbols in $C+H^{\mathrm{\infty }}$ on weighted Hardy spaces are independent of the weights

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