Spectrum of Rota-Baxter operators

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 ${\mathrm{r}\mathrm{b}}_{\lambda }(A)$ 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 $M_n(F)$, $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}F=0$: it is shown that ${\mathrm{r}\mathrm{b}}_{\lambda }(M_n(F))=2n-1$.