Spectrum of Rota-Baxter operators

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 ${\mathrm{rb}}_{\lambda }(M_n(F))=2n-1$.

arXiv comment23 p.; v2: Introduction is extended

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• Rota-Baxter operators of nonzero weight on the split Cayley-Dickson algebra
• Rota-Baxter operators of weight zero on the matrix algebra of order three without unit in kernel
• Monomial Rota-Baxter operators on free commutative non-unital algebra