Spectrum of Rota–Baxter operators

Rota–Baxter operators have been studied since 1960, and there are lots of their applications in mathematical physics, number theory, and noncommutative geometry. We state a surprisingly general property of such operators: the spectrum of every Rota–Baxter operator of weight [Formula: see text] on a finite-dimensional unital (not necessarily associative) algebra is a subset of [Formula: see text]. We even extend this result to the case of infinite-dimensional algebraic algebras (when [Formula: see text]). Based on these results, we define a new invariant of an algebra: the Rota–Baxter [Formula: see text]-index [Formula: see text] of an algebra A as the infimum of the degrees of minimal polynomials of all Rota–Baxter operators of weight [Formula: see text] on A. We compute the Rota–Baxter [Formula: see text]-index for the matrix algebra [Formula: see text], [Formula: see text]: it is shown that [Formula: see text].