We provide complete characterisations for the compactness of weighted composition operators between two distinct $L^{p}$ -spaces, where $1\leq p\leq \infty$ . As a corollary, when the underlying measure space is nonatomic, the only compact weighted composition map between $L^{p}$ -spaces is the zero operator.

This study adopted a quantitative approach to explore learning adjustment in terms of the students’ perceived adjustments in teaching, cognitive and social presence in the community of inquiry (CoI) framework from face-to-face classroom to online learning, and then from online to blended learning in the case of a higher education institution in Hong Kong. The online CoI survey was conducted to collect data from a convenience sample of 212 participating students who experienced the transition from classroom to online learning, and then the transition from online to blended learning in the case of the higher education institution. Analysis of variance, pairwise comparisons, correlation and multi-group analyses were performed on the collected data. The analytical results indicated that the students perceived four adjustments in their CoI presences in the online classroom and blended learning. (1) Classroom learning exhibited the strongest CoI presence while online learning rendered the weakest CoI presence. (2) Cognitive presence was the most important while social presence was the least important for learning. (3) All correlations among the CoI presence were positive with large correlations in online and blended learning modes. (4) All the teaching, cognitive and social adjustments were positively and largely correlated. These findings provide implications for further studies on exploring the explanation for the student’s learning adjustment and investigating the learning improvement strategies for online education.

Let $u$ and $\unicode[STIX]{x1D711}$ be two analytic functions on the unit disc $D$ such that $\unicode[STIX]{x1D711}(D)\subset D$ . A weighted composition operator $uC_{\unicode[STIX]{x1D711}}$ induced by $u$ and $\unicode[STIX]{x1D711}$ is defined by $uC_{\unicode[STIX]{x1D711}}f:=u\cdot f\circ \unicode[STIX]{x1D711}$ for every $f$ in $H^{p}$ , the Hardy space of $D$ . We investigate compactness of $uC_{\unicode[STIX]{x1D711}}$ on $H^{p}$ in terms of function-theoretic properties of $u$ and $\unicode[STIX]{x1D711}$ .

We investigate the boundedness, compactness, invertibility and Fredholmness of weighted composition operators between Lorentz spaces. It is also shown that the classes of Fredholm and invertible weighted composition maps between Lorentz spaces coincide when the underlying measure space is nonatomic.