Crouzeix's conjecture for new classes of matrices

06/21/2023

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by   Ryan O'Loughlin, et al.

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For a matrix A which satisfies Crouzeix's conjecture, we construct several new classes of matrices from A for which the conjecture will also hold. We discover a new link between cyclicity and Crouzeix's conjecture, which affirms the conjecture in the positive for a class of matrices. We pose several open questions, which if proved, will prove Crouzeix's conjecture. We also begin an investigation into Crouzeix's conjecture for symmetric matrices and in the case of 3 × 3 matrices, we show Crouzeix's conjecture holds for symmetric matrices if and only if it holds for analytic truncated Toeplitz operators.