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arXiv · 2006.02654

Spectrum of Rota-Baxter operators

Vsevolod Gubarev
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Abstract

We prove that the spectrum of every Rota-Baxter operator of weight $\lambda$ on a unital algebraic (not necessarily associative) algebra over a field of characteristic zero is a subset of $\{0,-\lambda\}$. For a finite-dimensional unital algebra the same statement is shown to hold without a restriction on the characteristic of the ground field. Based on these results, we define the Rota-Baxter $\lambda$-index $\mathrm{rb}_\lambda(A)$ of an algebra $A$ as the infimum of the degrees of minimal polynomials of all Rota-Baxter operators of weight $\lambda$ on $A$. We calculate the Rota-Baxter $\lambda$-index for the matrix algebra $M_n(F)$, $\mathrm{char}\,F = 0$: it is shown that $\mathrm{rb}_\lambda(M_n(F)) = 2n-1$.

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Vsevolod Gubarev. 2020-06-04. Spectrum of Rota-Baxter operators. https://doi.org/10.1142/s0218196725500195

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