Commutators of two compressed shifts and the Hardy space on the bidisc (Q402905)

For \(1 \leq p \leq \infty\), let \(H^p ( D )\) denote the one variable Hardy space on the open unit disc \(D\) in \(\mathbb{C}\) and \(H^p ( {D^2 } )\) denote the two variable Hardy space on \( D^2 = D \times D\). Let \(m\) be the normalized Lebesgue measure on \(T^2 = T \times T\), where \(T = \partial D\). Each \(f\) in \(H^2 ( {D^2 } )\) has a radial limit \(f^*\) defined on \(T^2\) a.e.\ and let \(H^p ( {T^2 } ) = \{ f^* :f \in H^p ( {D^2 } )\}\). Then \( H^p ( {T^2 } )\) is a Banach space in \(L^p ( {T^2 } ) = L^p ( {T^2 ,m} )\). It is known that \(H^p ( {D^2 } )\) is isometrically isomorphic to \(H^p ( {T^2 } )\). A closed subspace \(M\) of \(H^2 ( {D^2 } )\) is said to be an invariant subspace if \( zM \subseteq M\) and \(wM \subseteq M\). Let \(N\) be the orthogonal complement of \(M\) in \(H^2 ( {D^2 } )\). For a subset \(E\) of \({D^2 }\), let \(M = \{ f \in H^2 ( {D^2 } ):f = 0\;\text{on}\;E\}\). Then \(N\) is the closed linear span of \( \{ ( {1 - \overline a z} )^{ - 1} ( {1 - \overline b w} )^{ - 1} :( {a,b} ) \in E\}\). For a function \(\psi\) in \(H^2 ( {D^2 } )\), put \[ S_\psi f = P^N ( {\psi f} )\;\;( {f \in N \bigcap H^\infty ( {D^2 } )} ), \] where \(P^N\) is the orthogonal projection onto \(N\). The question whether there exists a function \(\phi\) in \(H^\infty ( {D^2 } )\) such that \( S_\psi = S_\phi\) when \(S_\psi\) is bounded has been answered negatively in [\textit{E. Amar} and \textit{C. Menini}, Bull. Lond. Math. Soc. 34, No. 3, 369--373 (2002; Zbl 1024.47005)]. {Problem 3.1} Let \(\psi\) be a function in \(H^2 ( {D^2 } )\). If \(S_\psi\) is of finite rank, then does there exist a function \(\phi\) in \({H^\infty ( {D^2 } )}\) such that \( S_\psi = S_\phi\)? The paper under the review solves this problem under some conditions. {Problem 3.2} If the Pick matrix \[ [( {1 - \eta _i \overline {\eta _j } } )k_{\zeta _j } ( {\zeta _i } )] \geq 0\;( {1 \leq j \leq n} ), \] then does there exist a function \(\phi\) in \({H^\infty ( {D^2 } )}\) such that \( \| \phi \|_\infty \leq 1\) and \(\phi ( {\zeta _j } ) = \eta _j\) (\({1 \leq j \leq n} \))? In the second problem, it is easy to see the converse is valid. It is known by [\textit{J. Agler} and \textit{J. E. McCarthy}, J. Reine Angew. Math. 506, 191--204 (1999; Zbl 0919.32003)] that the problem can in general be solved negatively. The paper under the review considers it in some special case.