Rank of truncated Toeplitz operators
A Toeplitz operator \(T_\phi\) with symbol \(\phi\in L^\infty\) is a map between Hardy spaces \(H^2\ni f\mapsto P(\phi f)\in H^2\), where \(P\) is the orthogonal projection onto \(H^2\). Recall that \(T_{\overline{f}g}=T_{\overline{f}}T_g\) for \(f,g\in H^\infty\). A~truncation to a subspace \(K_u^2=H^2\ominus u H^2\) with \(u\) an inner function is defined as \(A_\phi f=P_u(\phi f)\), with \(f\in K_u^2\cap H^\infty\) and \(P_u\) the projection onto \(K_u^2\). The paper characterizes the rank of the truncated operators in terms of \(u\). The associated Hankel operators \(H_\phi\) have an important role in deriving these results. For example, if \(K_u^2\) is infinite dimensional and \(\psi=\overline{u}r_1+ur_2\) with \(r_1\) and \(r_2\), rational functions, then \(\operatorname{rank}(A_\psi)=\alpha(\overline{r_1})+\alpha(r_2)\), where \(\alpha(r)\) denotes the number of poles of \(r\) inside the unit disk, which is also the rank of the Hankel operator \(H_r\). Also, the ranges of \(A_{\overline{u}r_1}\) and \(A_{ur_2}\) are given explicitly. Similar theorems are proved when \(u\) is a finite Blaschke product of degree \(n\). An example: if \(\phi=\overline{g}+f\), \(f\in H^2, g\in zH^2\), and \(\operatorname{rank}(A_f)+\operatorname{rank}(A_{\overline{g}})\leq n\), then \(\operatorname{rank}(A_\phi)=2n-\mathcal{Z}(u,f)-\mathcal{Z}(u,g)\), where \(\mathcal{Z}(a,b)\) is the number of common zeros of \(a\) and \(b\), and if \(\operatorname{rank}(A_f)+\operatorname{rank}(A_{\overline{g}})> n\), then \(\operatorname{rank}(A_\phi)\geq n-\min\{\operatorname{rank}(A_{\overline{g}}),\operatorname{rank}(A_f)\}\).
scientific article; zbMATH DE number 6706374