We give a classification of all linear natural operators transforming affinors on each n-dimensional manifold M into affinors on T A M, where T A is the product preserving bundle functor given by a Weil algebra A, under the condition that n ≥ 2. We recall that an affinor on a manifold M is a tensor field of type (1, 1) on M , which can be interpreted as a linear endomorphism of the tangent bundle T M . We will denote by aff(M ) the vector space of all affinors on M . Let A be a Weil algebra and

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... T A the Weil functor corresponding to A, which is a product preserving bundle functor (see [3] ). Fix also a positive integer n. A lifting of affinors to T A is, by definition, a system of maps Λ M : aff(M ) → aff(T A M ) indexed by n-dimensional manifolds and satisfying for all such manifolds M , N , for every embedding f : M → N and for all t ∈ aff(M ), u ∈ aff(N ) the following implication: 2000 Mathematics Subject Classification: 58A32, 58A20, 53A55. Key words and phrases: natural operator, affinor, product preserving bundle functor, Weil algebra. for V ∈ T T A M . A trivial verification shows that C is a linear lifting of affinors to T A . Clearly, this lifting is the composition of the complete lifting of affinors to affinors on the Weil bundle and a natural affinor on the Weil bundle. Example 2. Let L : A → A be an R-linear map. For every n-dimensional manifold M and every t ∈ aff(M ) we have the trace function tr t : M → R, and so T A tr t : is as in Example 1. A trivial verification shows that L is a linear lifting of affinors to T A . It is worth pointing out that this lifting is nothing but a sum of products of linear liftings of affinors to functions on the Weil bundle (see [5] ) and natural affinors on the Weil bundle. Example 3. Let D : A × A → A be an R-bilinear map with the property that D(P · Q, R) = P · D(Q, R) + D(P, R) · Q for P, Q, R ∈ A. For every n-dimensional manifold M and every t ∈ aff(M ) we have the map d(T A tr t) : T T A M → A, which is the exterior derivative of T A tr t. Clearly, for every V ∈ T T A M the map r t,V : A P → D(P, d(T A tr t)(V )) ∈ A is a differentiation of the algebra A. It is well known that every differentiation of the Weil algebra A determines in a natural way a vector field on T A N for each manifold N (see [2] for a construction of such natural vector fields). Denote by r t,V M the vector field on T A M determined by r t,V . Define for V ∈ T T A M . A trivial verification shows that D is a linear lifting of affinors to T A . Observe that this lifting is nothing but a sum of tensor products of natural vector fields on the Weil bundle and linear liftings of affinors to 1-forms on the Weil bundle (see [5] ). We are now in a position to formulate our main result. Theorem. If n ≥ 2 then for each linear lifting Λ of affinors to T A there are C ∈ A, an R-linear map L : A → A and an R-bilinear map D : A×A → A with the property that D(P ·Q, R) = P ·D(Q, R)+D(P, R)·Q for P, Q, R ∈ A, such that Λ = C + L + D. Moreover , C, L and D are uniquely determined.