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.)
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