On the essential norms of Toeplitz operators with continuous symbols

Abstract

It is well known that the essential norm of a Toeplitz operator on the Hardy space H p (T), 1 < p < ∞ is greater than or equal to the L ∞ (T) norm of its symbol. In 1988, A. Böttcher, N. Krupnik, and B. Silbermann posed a question on whether or not the equality holds in the case of continuous symbols. We answer this question in the negative. On the other hand, we show that the essential norm of a Toeplitz operator with a continuous symbol is less than or equal to twice the L ∞ (T) norm of the symbol and prove more precise p-dependent estimates.

Sharp estimates for conditionally centred moments and for compact operators on $L^p$ spaces

“…If one is not interested in sharp constants, then it usually does not matter whether one knows m(X) or M(X). However, the latter appears naturally in estimates for the essential norms of operators by their measures of noncompactnes and it is desirable to know the value of M(X) (see [3], [14], [19], [32]). It is well known that m(L p ([0, 1])) = 1, 1 ≤ p < ∞ (see, e.g., [26,Lemma 19.3.5]).…”