关于具有交换系数的奇异铅笔
On singular pencils with commuting coefficients
AI总结：
本文研究矩阵多项式谱与系数Taylor谱的关系，证明奇异多项式等价于零点在系数Taylor谱中，但该等价在无穷维算子情形失效，并给出新谱描述及Kronecker形式。
摘要：
我们研究了矩阵（或算子）多项式的谱与其系数的Taylor谱之间的关系。我们证明了交换矩阵的多项式是奇异的，即其谱为整个复平面，当且仅当(0, 0, ... , 0)属于其系数的Taylor谱。另一方面，我们证明了如果考虑无穷维Hilbert空间上的算子作为多项式的系数，这一等价关系不再成立。作为推论，我们提出了k元矩阵组的(Taylor)谱的一种新描述，并否定了文献中先前提出的猜想。此外，我们指出了具有交换系数的铅笔的Kronecker形式。
英文摘要：
We investigate the relation between the spectrum of matrix (or operator) polynomials and the Taylor spectrum of its coefficients. We prove that the polynomial of commuting matrices is singular, i.e. its spectrum is the whole complex plane, if and only if (0, 0, ... , 0) belongs to the Taylor spectrum of its coefficients. On the other hand we prove that this equivalence is not longer true if we consider the operators on infinite dimensional Hilbert space as coefficients of polynomial. As a consequence we could propose a new description of (Taylor) spectrum of k-tuple of matrices and we could disprove the conjecture previously proposed in the literature. Additionally, we pointed out the Kronecker forms of the pencils with commuting coefficients.
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