The classical interpolation problem on the disk $/mathbb{D}$ is: given $n$ distinct points $z_1,/ldots,z_n$ in $/mathbb{D}$ and $n$ complex numbers $w_1,/ldots,w_n$, find the function $f$ in $H^{/infty}(/mathbb{D})$ with the smallest norm. In 1967, Donald Sarason gave an operator theory approach to this problem by determining the commutants of certain operator on the model space, which is called the generalized interpolation. Sz.Nagy and Foias extended Sarason's theorem as their commutant lifting theorem for a contraction   on an abstract Hilbert space. In this talk, we present a generalized interpolation for compressed shifts on the Hardy space on the bidisk. We also consider the Nevanlinna-Pick type interpolation and Carath/'{e}odory type interpolation in two variables.

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