Toeplitz operators between distinct Bergman spaces

Abstract

For $-1<α<\infty$, let $ω_α(z)=(1+α)(1-|z|^2)^α$ be the standard weight on the unit disk. In this note, we provide descriptions of the boundedness and compactness for the Toeplitz operators $T_{μ,β}$ between distinct weighted Bergman spaces $L_{a}^{p}(ω_α)$ and $L_{a}^{q}(ω_β)$ when $0<p\leq1$, $q=1$, $-1<α,β<\infty$ and $0<p\leq 1<q<\infty, -1<β\leqα<\infty$, respectively. Our results can be viewed as extensions of Pau and Zhao's work in \cite{Pau}. Moreover, partial of main results are new even in the unweighted settings.