# Journal of Operator Theory

Volume 79, Issue 1, Winter 2018 pp. 225-265.

Compressions of the shift on the bidisk and their numerical ranges**Authors**: Kelly Bickel (1) and Pamela Gorkin (2)

**Author institution:**(1) Department of Mathematics, Bucknell University, 380 Olin Science Building, Lewisburg, PA 17837, U.S.A.

(2) Department of Mathematics, Bucknell University, 380 Olin Science Building, Lewisburg, PA 17837, U.S.A.

**Summary:**We consider two-variable model spaces associated to rational inner functions on the bidisk, which always possess canonical $z_2$-invariant subspaces $\mathcal{S}_2$, and study the compression of multiplication by $z_1$ to $\mathcal{S}_2$, namely $ S^1_{\Theta}:= P_{\mathcal{S}_2} M_{z_1} |_{\mathcal{S}_2}$. We show that these compressed shifts are unitarily equivalent to matrix-valued Toeplitz operators with nice symbols and characterize their numerical ranges and radii. We later specialize to particularly simple rational inner functions and study the geometric properties of the associated numerical ranges.

**DOI:**http://dx.doi.org/10.7900/jot.2017feb14.2143

**Keywords:**compressions of the shift, numerical range, inner function, bidisk

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