Abstract. Let $[z-w]$ be the smallest invariant subspace of $H^2$ over the bidisk containing $z-w$. Let $M$ be an invariant subspace satisfying $[z-w]\subsetneqq M \subset H^2$. We denote by $F^M_z$ the compression operator of the multiplication operator by $z$ on $M\ominus w M$ which is called the fringe operator of $M$. It is proved that $F^M_z$ is Fredholm and ${\rm ind} F^M_z=-1$. Its generalizations are also given.

DOI: 10.14232/actasm-017-012-6

AMS Subject Classification (1991): 47A15, 32A35, 47B35

Keyword(s): Hardy space over the bidisk, invariant subspace, fringe operator, Fredholm operator, Fredholm index, Bergman space

Received February 4, 2017, and in revised form April 7, 2017. (Registered under 12/2017.)