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It is well known that the essential norm of a Toeplitz operator on the Hardyspace $H^p(\mathbb{T})$, $1 < p < \infty$ is greater than or equal to the$L^\infty(\mathbb{T})$ norm of its symbol. In 1988, A. B\"ottcher, N. Krupnik,and B. Silbermann posed a question on whether or not the equality holds in thecase of continuous symbols. We answer this question in the negative. On theother hand, we show that the essential norm of a Toeplitz operator with acontinuous symbol is less than or equal to twice the $L^\infty(\mathbb{T})$norm of the symbol and prove more precise $p$-dependent estimates.

On the essential norms of Toeplitz operators with continuous symbols