Generalized interpolation for type 1 subdiagonal algebras

Let \({\mathfrak {A}}\) $\mathfrak{A}$ be a maximal subdiagonal algebra with diagonal \({\mathfrak {D}}\) $\mathfrak{D}$ in a \(\sigma \) $\sigma$ -finite von Neumann algebra \({\mathcal {M}}\) $\mathcal{M}$ with respect to a faithful normal conditional expectation \(\Phi \) $\mathrm{\Phi }$ . We firstly give a type decomposition of an invariant subspace \({\mathfrak {M}}\) $\mathfrak{M}$ of \({\mathfrak {A}}\) $\mathfrak{A}$ in the acting Hilbert space. We then revisit certain useful properties of type 1 subdiagonal algebras. It is shown that a two-sided invariant subspace \({\mathfrak {M}}\) $\mathfrak{M}$ in the noncommutative \(H^2\) $H^2$ space has the form \({\mathfrak {M}}=\oplus _{n\ge 1}^{col}W_nH^2\) $\mathfrak{M}={\oplus }_{n\ge 1}^{\mathit{col}}{W}_n{H}^2$ for a family of partial isometries \(\{W_n:n\ge 1\}\) $\{W_n:n\ge 1\}$ satisfying \( W_n^*W_m=0\) $W_n^{\ast }{W}_m=0$ when \(n\not =m\) $n\ne m$ , \(W_n^*W_n\in {\mathfrak {D}}\) $W_n^{\ast }{W}_n\in \mathfrak{D}$ and \(\sum _{n\ge 1} W_nW_n^*=I\) ${\sum }_{n\ge 1}{W}_n{W}_n^{\ast }=I$ if \({\mathfrak {D}}\) $\mathfrak{D}$ is a factor. Furthermore, we give a noncommutative version of the Sarason’s generalized interpolation theorem for such a two-sided invariant subspace of a type 1 subdiagonal algebra.