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The reducibility of compressed shifts on a class of quotient modules over the bidisk. (English) Zbl 1477.47027

It is shown that, for the rational inner function \(\theta(z,w)=\frac{zw+\bar{b}w+\bar{c}z+\bar{d}}{1+bz+cw+dzw}\), the compressed shift \(S_z=P_\theta T_z|_{\mathcal{K}_\theta}\) is reducible on the quotient module \(\mathcal{K}_\theta=H^2\ominus \theta H^2\) over the bidisk if and only if \(\theta\) is the product of two one-variable inner functions.

MSC:

47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47A15 Invariant subspaces of linear operators
32A10 Holomorphic functions of several complex variables
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
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References:

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