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On singular pencils with commuting coefficients

Vadym KovalPatryk Pagacz
Feb 2024
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摘要原文
We investigate the relation between the spectrum of a linear pencil and the Taylor spectrum of its coefficients. We prove that the linear pencil of commuting matrices is singular, i.e. its spectrum is the whole complex plane, if and only if (0, 0) belongs to the Taylor spectrum of its coefficients. This result implies two descriptions of Taylor spectrum of a pair of matrices. On the other hand we prove that this equivalence is not longer true if we consider the operators on infinite dimensional Hilbert space. Additionally, we pointed out the Kronecker forms of the pencils with commuting coefficients
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