We classify all Rota Baxter operators of nonzero weight on the matrix algebra of order three over an algebraically closed field of characteristic zero which are not arisen from the decompositions of the entire algebra into a direct vector space sum of two subalgebras.Keywords: Rota Baxter operator, matrix algebra.
RB-operators on M n (F )Over a field F , denote the algebra of all upper and lower triangular matrices of order n by U n and L n respectively.Example 1 [7]. Decomposing M n (F ) = L n ⊕ D n ⊕ U n (as vector spaces) and given an RB-operator defined on D n , we have a triangular-splitting RB-operator defined withDecomposing M n (F ) = M 1 ⊕M 2 ⊕M 3 (as vector spaces) and given an RB-operator defined on M 1 , we have a triangular-splitting RB-operator on M n (F ) defined with A + = M 2 ,Example 3. Let an algebra A be equal a direct vector space sum A − ⊕ A 0 ⊕ A + of its three subalgebras satisfying the conditions A 0 A − , A − A 0 ⊂ A + . Suppose that P is an RBoperator of weight 1 on A 0 . Then a linear operator R defined on A as R(a − + a 0 + a + ) = P (a 0 ) − a + is an RB-operator of weight 1 if R(A 0 )A − , A − R(A 0 ) ⊂ A − .