Essential norms of weighted composition operators and Aleksandrov measures
The purpose of this interesting paper is to improve earlier results by \textit{D. Sarason} [Lect. Notes Pure Appl. Math. 122, 545--565 (1990; Zbl 0712.47026)], \textit{J. A. Cima} and \textit{A. L. Matheson} [Pac. J. Math. 179, No.~1, 59--64 (1997; Zbl 0871.47027)] and by \textit{T. Kriete} and \textit{J. Moorhouse} [Trans. Am. Math. Soc. 359, No.~6, 2915--2944 (2007; Zbl 1115.47023)] and to present quantitative refinements that establish exact formulas for the essential and weak essential norms of weighted composition operators \(M_wC_\varphi\), acting either on the spaces \(M(\mathbb{T})\) and \(L^p(\mathbb{T})\), \(1 \leq p \leq \infty\), of complex Borel measures and \(p\)-integrable functions on the unit circle \(\mathbb{T}\), or on the Hardy space \(H^2\). Here, \(\varphi\) is an analytic self-map on the unit disc \(\mathbb{D}\) and \(w\) is a bounded Borel measurable function \(w:\mathbb{T} \rightarrow \mathbb{C}\) in the case of \(M(\mathbb{T})\) and \(w \in H^\infty\) in that of \(H^2\), under the assumption, in both cases, that \(|w|\) is continuous in the union of the closed supports of the Aleksandrov measures of \(\varphi\). The proofs are based on a general method of producing compact and weakly compact approximants for continuous linear operators \(T:L^1(X,\mu) \rightarrow L^1(Y,\nu)\) between spaces of integrable functions on metric measure spaces. The idea is originally due to Weiss and is reformulated in Section 2. Interpolation arguments and suitable test functions are also utilized in the proofs. A quantitative version of a result by \textit{E. Saksman} and the author in [J. Math. Anal. Appl. 298, No.~2, 501--522 (2004; Zbl 1072.47021)] on the (weak) essential norm of differences of composition operators is presented in Section 5. A nice example uses the obtained formula to derive explicit lower bounds of the essential norm of the difference.
scientific article; zbMATH DE number 5925578
scientific article