For a matrix $A$ which satisfies Crouzeix's conjecture, we construct several classes of matrices from $A$ for which the conjecture will also hold. We discover a new link between cyclicity and Crouzeix's conjecture, which shows that Crouzeix's Conjecture holds in full generality if and only if it holds for the differentiation operator on a class of analytic functions. 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 $3 \times 3$ matrices, we show Crouzeix's conjecture holds for symmetric matrices if and only if it holds for analytic truncated Toeplitz operators.

翻译：对于满足Crouzeix猜想的矩阵$A$，我们构造了若干类从$A$出发的矩阵，使得该猜想对它们同样成立。我们发现了循环性与Crouzeix猜想之间的新联系，表明Crouzeix猜想在完全一般情形下成立当且仅当它对某类解析函数上的微分算子成立。我们提出若干未解决问题，若得证则将证明Crouzeix猜想。我们还开始研究对称矩阵情形下的Crouzeix猜想，并在$3 \times 3$矩阵情形下证明：Crouzeix猜想对对称矩阵成立当且仅当它对解析截断Toeplitz算子成立。