Geodesic
Area differences under analytic maps and operators
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 3, pp. 817-838

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Motivated by the relationship between the area of the image of the unit disk under a holomorphic mapping $h$ and that of $zh$, we study various $L^2$ norms for $T_{\varphi }(h)$, where $T_{\varphi }$ is the Toeplitz operator with symbol $\varphi $. In Theorem \ref {thm:Transitivity}, given polynomials $p$ and $q$ we find a symbol $\varphi $ such that $T_{\varphi }(p)=q$. We extend some of our results to the polydisc.
Motivated by the relationship between the area of the image of the unit disk under a holomorphic mapping $h$ and that of $zh$, we study various $L^2$ norms for $T_{\varphi }(h)$, where $T_{\varphi }$ is the Toeplitz operator with symbol $\varphi $. In Theorem \ref {thm:Transitivity}, given polynomials $p$ and $q$ we find a symbol $\varphi $ such that $T_{\varphi }(p)=q$. We extend some of our results to the polydisc.
DOI : 10.21136/CMJ.2024.0023-24
Classification : 30H05, 30J99, 32A36, 47B35
Keywords: unit disk; polydisc; polynomial; Toeplitz operator; Bergman projection 
Çelik, Mehmet; Duane-Tessier, Luke; Marcial Rodriguez, Ashley; Rodriguez, Daniel; Shaw, Aden. Area differences under analytic maps and operators. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 3, pp. 817-838. doi: 10.21136/CMJ.2024.0023-24
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     journal = {Czechoslovak Mathematical Journal},
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     year = {2024},
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     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2024.0023-24/}
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