On positivity preservers with constant coeﬃcients and their generators

In this work we study positivity preservers T : R [ x 1 , ..., x n ] → R [ x 1 , ..., x n ] with constant coeﬃcients and deﬁne their generators A if they exist, i.e., exp( A ) = T . We use the theory of regular Fréchet Lie groups to show the ﬁrst main result. A positivity preserver with constant coeﬃcients has a generator if and only if it is represented by an inﬁnitely divisible measure (Main Theorem 4.7). In the second main result (Main Theorem 4.11) we use the Lévy–Khinchin formula to fully characterize the generators of positivity preservers with constant coeﬃcients. © 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).


Introduction
Non-negative polynomials are widely applied and studied.A first step to investigate non-negative polynomials is to study the convex cone Pos(K) := {p ∈ R[x 1 , . . ., x n ] | p(x) ≥ 0 for all x ∈ K} of non-negative polynomials on some K ⊆ R n .
Lemma 1.1 (folklore, see e.g.[16,Lem. 2.3]).Let n ∈ N 0 and let T : R[x 1 , . . ., x n ] → R[x 1 , . . ., x n ] be linear.Then for all α ∈ N n 0 there exist unique q α ∈ R[x 1 , . . ., x n ] such that The map T : R[x 1 , . . ., x n ] → R[x 1 , . . ., x n ] is called to have constant coefficients if q α ∈ R for all α ∈ N n 0 .The subset with q 0 = 1 is denoted by Positivity preservers are fully described by their polynomial coefficients.A sequence s = (s α ) α∈N n 0 is called a moment sequence if there exists a measure μ such that s α = x α dμ(x) holds for all α ∈ N n 0 .We have the following.
Example 1.3.Let n = 1.Then is a positivity preserver for all t ≥ 0. • The following is a linear operator T which is not a positivity preserver.
Example 1.4.Let k ≥ 3 and a ∈ R \ {0}.Then exp(a∂ k x ) := j∈N 0 is not a positivity preserver since q 2k = a 2 2 = 0 but q 2k+2 = 0, i.e., (j! • q j ) j∈N 0 is not a moment sequence.• From elementary calculations and measure theory we get the following additional properties of positivity preservers.Let T be a positivity preserver with constant coefficients and μ be a representing measure of the corresponding moment sequence s = (s α ) α∈N n 0 = (α!• q α ) α∈N n 0 , then for all f ∈ R[x 1 , . . ., x n ].For short we call μ also a representing measure of T .It follows for two such operators T and T with representing measures μ and μ that where μ * μ is the convolution of the two measures μ and μ : for any A ∈ B(R n ), see e.g.[3,Sect. 3.9].Here, χ A is the characteristic function of the Borel set A. For the supports we have supp (μ * μ ) = supp μ + supp μ . ( This can easily be proved from (2) or found in the literature, see e.g.[7,Prop. 14.5 (ii)] for a special case.We abbreviate for all k ∈ N and set μ * 0 := δ 0 with δ 0 the Dirac measure centered at 0. Previously we investigated the heat semi-group (see Example 1.3 for n = 1) and its action on (non-negative) polynomials, see [5,6].For n ∈ N we have that exp(tΔ)p 0 is the unique solution of the polynomial valued heat equation In [6] we observed the very strange behavior that several non-negative polynomials which are not sums of squares (e.g. the Motzkin and the Choi-Lam polynomial) become a sum of squares in finite time under the heat equation.In [6,Thm. 3.20] we showed that every non-negative p ∈ R[x, y, z] ≤4 becomes a sum of squares in finite time.All these observations hold for the heat equation, i.e., the family (exp(tΔ)) t≥0 of positivity preservers with constant coefficients with the generator . So a natural question to further study these effects is to understand any family (exp(tA)) t≥0 of positivity preservers and hence to determine all possible generators A. This is the main question of the current work for the constant coefficient case.It is answered in Main Theorem 4.11.
The paper is structured as follows.To define and work with exp(A) we repeat in the preliminaries for the readers convenience the notion of Lie groups, Lie algebras, and especially their lesser known infinite dimensional versions of regular Fréchet Lie groups.We will see that D is a regular Fréchet Lie group with a Lie algebra d.By Theorem 1.2 we transport this property to sequences in Section 3. Section 4 contains the two main results.In the first Main Theorem 4.7 we show that a positivity preserver with constant coefficients has a generator if and only if it is represented by an infinitely divisible measure.Using the Lévy-Khinchin formula (which is also given in the preliminaries, see Theorem 2.18) we give in the second Main Theorem 4.11 the full description of all generators of positivity preservers with constant coefficients.In Section 5 we discuss a strange action on non-negative polynomials on [0, ∞) caused by the heat equation with Dirichlet boundary conditions.We end this paper with a summary and an open question.

Preliminaries
Lie groups and Lie algebras are standard concepts in mathematics [28].However, this only applies to the finite dimensional cases.For the readers convenience we give here the infinite dimensional definitions and examples we need to make the paper as self-contained as possible.For the sake of completeness we also include the explicit statement of the Lévy-Khinchin formula in Theorem 2.18.

Lie groups and their Lie algebras
The connection between the Lie algebra g and the Lie group G is given by the exponential map exp : g → G.For the special case G = Gl(m, C), m ∈ N, the exponential mapping fulfills the following.Lemma 2.2 (see e.g.[28,p. 134,Ex. 15]).Let m ∈ N. The following hold: The proof of Lemma 2.2 (ii) follows from formal power series calculations.For more on Lie groups and Lie algebras see e.g.[28].
The charts ϕ : U ⊆ R n → G of the n-dimensional smooth manifold G induce the Euclidean topology on the group G and (A, B) → AB −1 is smooth with respect to this topology.When extending this to infinite dimensions more than one topology is possible and the choice of topology on G is important.

Fréchet spaces
Definition 2.3.A topological vector space V is called a Fréchet space if the following three conditions are fulfilled: equipped with the topology induced by the semi-norms In other words we have the convergence • For more on topological vector spaces see e.g.[27,26].

Regular Fréchet Lie groups and their Lie algebras
Already Hideki Omori stated the following, see [17, pp.III-IV]: [G]eneral Fréchet manifolds are very difficult to treat.For instance, there are some difficulties in the definition of tangent bundles, hence in the definition of the concept of C ∞ -mappings.Of course, there is neither an implicit function theorem nor a Frobenius theorem in general.Thus, it is difficult to give a theory of general Fréchet Lie groups.
A more detailed study is given by Omori in [18] and the theory successfully evolved since then, see e.g.[17,12,18,25,29,23] and references therein.We will give here only the basic definitions which will be needed for our study.

The Lie group D
holds for all d ∈ N 0 and A ∈ D the D d are well-defined.From Definition 2.6 we see that D d consists only of operators of the form as usual for (unbounded) operators [21].Definition 2.6 can then even be used to calculate We will now see that D 3 is even a commutative group.For that it is sufficient to find for any i.e., every A ∈ D 3 has the unique inverse We have seen in the previous example that (D d , • ) for n = 1 and d = 3 is a commutative group.This holds for all n ∈ N and d i.e., the system ( 5) of equations has a unique solution gained by induction.Hence, for every A ∈ D d there exists a unique B ∈ D d with AB = BA = 1.
From Lemma 2.9 we have seen that (D d , • ) for any n ∈ N and d ∈ N 0 is a commutative group.Let be an affine linear map.Then ι d in ( 7) is a diffeomorphism and it is a coordinate map for Hence, the map ι d shows the following.Proof.The map ι d in ( 7) is a diffeomorphism between D d and {1} × R ( n+d n )−1 .Hence, D d is a differentiable manifold which possesses the group structure (D d , • ).By ( 5) and (6) we have that the map

The Lie algebra d d of D d
Since every A ∈ D d is a linear map between finite-dimensional vector spaces we can choose a basis of R[x 1 , . . ., x n ] ≤d and get a matrix representation Ã of A. Take the monomial basis of R[x 1 , . . ., x n ] ≤d .Then Ã is an upper triangular matrix with diagonal entries 1.

Example 2.11 (Example 2.8 continued). Let n = 1 and d = 3. Then every
and we therefore set Hence, ( Ã − id) 4 = 0 as a matrix and also (A − 1) 4 = 0 as an operator on R[x] ≤3 .From Lemma 2.2 we find that the matrix valued exponential map Since also (A − 1) 4 = 0 for all A ∈ D 3 we can use ( 9) also for the differential operators in D 3 : To determine the image log D 3 recall that also log is an injective map by Lemma 2.2 and hence log D 3 is 3-dimensional with d 0 = 0, i.e., we have log In summary, since A 4 = 0 for all A ∈ d 3 we have that exp : is surjective with inverse log : Therefore, d 3 is the Lie algebra of D 3 and exp in (10) is the exponential map between the Lie algebra d 3 and its Lie group D 3 with inverse log in (11).
The previous example of the Lie algebra d 3 of the Lie group D 3 holds for all n ∈ N and d ∈ N 0 .We define the following.Definition 2.12.Let n ∈ N and d ∈ N 0 .We define x n ] ≤d and we have the following.
Proof.Follows from Lemma 2.2 similar to Example 2.11.

The regular Fréchet Lie group D and its Lie algebra d
In Section 2.4 and 2.5 we have seen that (D d , • ) is a Lie group with Lie algebra (d d , • , +) for all d ∈ N 0 .Hence, similar to Definition 2.12 we define the following.Definition 2.14.Let n ∈ N. We define For D and d we have the following.
Theorem 2.15.Let n ∈ N. Then the following hold: (iii): At first we show that exp : d → D is well-defined.To see this note that for any A ∈ d we have i.e., A k contains no differential operators of order ≤ k − 1.Hence, the sum It is easy to see that d and D are both infinite dimensional smooth (Fréchet) manifolds.Hence, summing everything up we have the following.
Theorem 2.17.Let n ∈ N. Then (D, • ) as a Fréchet space is a commutative regular Fréchet Lie group with the commutative Fréchet Lie algebra (d, • , +).The exponential map is smooth and bijective with the smooth and bijective inverse Proof.We have that D is an infinite dimensional smooth manifold, D is a Fréchet space (with the coefficient-wise convergence topology, Example 2. In the previous proof we can also replace (12) by the fact that a function
Theorem 2.18 (Lévy-Khinchin, see e.g.[13,Cor. 15.8] or [14,Satz 16.17]).Let n ∈ N and μ be a measure on R n .The following are equivalent: (i) μ is infinitely divisible.(ii) There exist a vector b ∈ R n , a symmetric matrix Σ ∈ R n×n with Σ 0, and a measure ν on R n such that holds for the characteristic function of μ.

The regular Fréchet Lie group structure of sequences
We define appropriate sets S, s R N n 0 of sequences with convolution * .From these definitions we will see that the set of sequences R N n 0 , and therefore also S and s, inherit the Fréchet topology Define the linear and bijective map and on R N n 0 the convolution * as * : R We abbreviate The map D in ( 13) is of course a Fréchet space isomorphism.The map D also equips S and s with the Fréchet topology of D and d, respectively.In summary, we have the following.→ (d, • , +).Apply both isomorphisms to Theorem 2.17 and the assertion is proved.
We want to point out that Hirschman and Widder [11] extensively investigated the inversion theory of convolution of measures and functions.But from Theorem 3.3 we see that the convolution of their moments is trivial when we allow signed moment sequences.
For * on S we find from (2) the following.Note, that every sequence s ∈ R N n 0 is represented by a signed measure, see e.g.[19,2,24,20].It is even possible to restrict the support to [0, ∞) n , see e.g.[2,24] or to use only linear combinations of Dirac measures [1].
Because of the connection of positivity preservers to moment sequences by Theorem 1.2 we define the following.Definition 3.5.Let n ∈ N. We define Then S + is a base of the moment cone.α ) α∈N n 0 ∈ S + for all n ∈ N 0 be such that s (n) → s in the Fréchet topology (Example 2.4), i.e., s , by Haviland's Theorem [10] L s is a moment functional and s ∈ S + .
(ii): Since s, t ∈ S + we have that s is represented by μ and t is represented by ν.Therefore, s * t is represented by μ * ν by Corollary 3.4.
The following is another trivial consequence of (2).A moment sequence is called indeterminate if it has two and therefore infinitely many representing measures.Corollary 3.8.Let n ∈ N and s, t ∈ S + .If s or t is indeterminate then s * t ∈ S + is indeterminate.

Generators of positivity preservers with constant coefficients
In Section 2.6 we showed that D is a Fréchet Lie group with Fréchet Lie algebra d.With the smooth and bijective exp : d → D with the inverse log : D → d we can now easily go from positivity preservers with constant coefficients to their generators.From Theorem 1.2 we see that D| S + : S + → D + in ( 13) is an isomorphism.
Examples 4.2.Let n ∈ N. Then we have the following: The following result shows that the cases in Examples 4.2 are the only generators of positivity preservers of finite rank.
Proof.Let k ≥ 3 and a k = 1.By Example 1.4 (and 4.3) we have that exp(∂ k x ) ∈ D + , i.e., it is not a positivity preserver.Hence, there exists a Assume to the contrary that A ∈ d + .By scaling x and A we have that holds for all λ > 0. By Theorem 2.17 we have that [exp for any initial value p 0 ∈ Pos(R n ) fulfills p t ∈ Pos(R n ) for all t ≥ 0.
Proof.Since p t = exp(tA)p 0 is the unique solution of the time evolution (15) we have that (i) ⇔ exp(tA) is a positivity preserver for all t ≥ 0 ⇔ (ii).
While we have d + ⊆ log D + equality does not hold as we will see in Corollary 4.10.The existence of a positivity preserver is equivalent to the existence of an infinitely divisible representing measure as the following result shows.It is known that the only infinitely divisible measures with compact support are δ x for x ∈ R n , see e.g.[14, p. 316].Therefore, we have that μ is not infinitely divisible and hence log A ∈ d + .•

Main
The previous example implies that the inclusion d + ⊆ log D + is proper.
Proof.By Main Theorem 4.7 "(i) ⇔ (ii)" we have that (i) A ∈ d + if and only if exp A has an infinitely divisible representing measure μ, i.e., by Theorem 1.2 we have By Theorem 2.17 we can take the logarithm and hence ( 16) is equivalent to With the isomorphism we have that ( 17) is equivalent to But the right hand side of ( 18) is now the characteristic function = log e itx dμ(x) of μ.Hence, by the Lévy-Khinchin formula (see Theorem 2.18) we have After a formal power series expansion of e itx in the Fréchet topology of C[[x 1 , . . ., x n ]], see Example 2.4, and a comparison of coefficients we have that ( 19) is equivalent to (ii) which ends the proof.
From the previous result we see that the difference between d + and a moment sequence is that the representing (Lévy) measure ν in (19) can have a singularity of order ≤ 2 at the origin.

A strange action on Pos([0, ∞))
Before we end this work we want to discuss an example of a strange positivity action on [0, ∞).We take the following example.
for all x, y ≥ 0 and t > 0, see [8,Exm. 4 ) for all t > 0 and hence we look at the time-dependent moments x j • k t (x, y) • u 0 (y) dx dy for all j ∈ N 0 and t > 0.
The action on R[x] is then given by (T t p)(x) = ∞ 0 p(y) • k t (x, y) dy.We find for p = 1 that for any t ∈ (0, ∞).While for p = x j with j ∈ N we get for the time-dependent moments by partial integration These are the recursive relations we already encountered with the heat equation on R, see [5,6].Therefore, for odd polynomials p ∈ R[x] we have does not mean that the positivity preservers are exp(t∂ 2 x ).The domain and the corresponding boundary conditions have to be taken into account as is done and is well-known in the partial differential equation literature and semi-group theory.Besides the Open Problem 6.1 below this is another direction where further studies have to be done.The constant coefficient case is completely solved by Main Theorem 4.11.Part of the non-constant coefficient cases are covered by the previous case that A is a degree preserving positivity preserver.For the non-constant coefficient cases note that we are then in the framework of a non-commutative infinite dimensional Lie group and its Lie algebra.Here the theory is much richer, see e.g.[17,12,18,25,29,23] and references therein.

Funding
The author and this project are supported by the Deutsche Forschungsgemeinschaft DFG with the grant DI-2780/2-1 and his research fellowship at the Zukunfskolleg of the University of Konstanz, funded as part of the Excellence Strategy of the German Federal and State Government.

(
iv) The Fréchet Lie algebra g of G is isomorphic to the tangent space T e G of G at the unit element e ∈ G. (v) exp : g → G is a smooth mapping such that d dt t=0 exp(tu) = u holds for all u ∈ g. (vi) The space C 1 (G, g) of C 1 -curves in G coincides with the set of all C 1 -curves in G under the Fréchet topology.

Remark 2 . 7 .
We can also define D d by D/ ∂ α | |α| = d + 1 .Both definitions are almost identical.However, Definition 2.6 has the following advantage.In D/ ∂ α | |α| = d + 1 we have the problem that we are working with equivalence classes and hence we can not calculate A + B for A ∈ D d and B ∈ D e for d = e.With Definition 2.6 we can calculate A + B for A ∈ D d and B ∈ D e with d = e since A + B is defined on dom n. Then (5) can be solved by induction on |γ|.For |γ| = 0 we have c 0 = a 0 • b 0 and a 0 = c 0 = 1, i.e., b 0 = 1.So assume (5) is solved for all c γ with |γ| ≤ d − 1.Then for any γ ∈ N 0 with |γ| = d we have

Theorem 2 . 13 .
Let n ∈ N and d ∈ N 0 .Then (d d , • , +) is the Lie algebra of the Lie group (D d , • ) with exponential map exp : The maps exp : d → D and log : D → d are inverse to each other.Proof.(i): That is clear.(ii): That A •B = B •A holds for all A, B ∈ D is clear.The inverse of A ∈ D is uniquely determined by solving (5) to get (6) for all γ ∈ N n 0 .This is a formal power series argument with coordinate-wise convergence (as in the Fréchet topology, Example 2.4).

( 1 −
in the Fréchet topology of D R[[∂ 1 , . . ., ∂ n ]] ∼ = R[[x 1 , . . ., x n ]], see Example 2.4.In other words, the coefficients c α depend only on A k for k = 0, . . ., |α|, we therefore have c K,α = c α for all K > |α|, and hence exp A ∈ D is well-defined.With that we have exp d ⊆ D. For equality we give the inverse map in (v).(iv): To show that log : D → d is well-defined the same argument as in (iii) holds for (1 − A) k with A ∈ D. It shows that log A ∈ d for all A ∈ D is well-defined and we have log D ⊆ d.(v):To prove that exp and log are inverse to each other we remarkR[x 1 , . . ., x n ] = d∈N 0 R[x 1 , . . ., x n ] ≤d .For d ∈ N 0 define exp d A := d k=0 A k k! and log d A := − d k=1 A) k k .Then for every p ∈ R[x 1 , . . ., x n ] with d = deg p we have exp(log A)p = exp d (log d A)p = Ap for all A ∈ d by Theorem 2.13, i.e., exp(log A) = A for all A ∈ D. Similarly, we have log(exp A)p = Ap for all A ∈ d.This also shows the remaining assertions exp(d) = D and log D = d from (iii) and (iv).For R[[∂ 1 , . . ., ∂ n ]] ⊇ d, D being a Fréchet space with the before mentioned topology (of coordinate-wise convergence, Example 2.4) we have the following.Corollary 2.16.Let n ∈ N and let d, D ⊆ R[[∂ 1 , . . ., ∂ n ]] be Fréchet spaces (equipped with the coordinate-wise convergence).Then the following hold: (i) D × D → D, (A, B) → AB −1 is smooth.(ii) exp : d → D is smooth and d dt t=0 exp(tu) = u holds for all u ∈ g. (iii) log : d → D is smooth.Proof.(i): Let A = α∈N n 0 a α ∂ α and B = α∈N n 0 b α ∂ α with a 0 = b 0 = 1.From (5) we see that the multiplication is smooth since every coordinate c γ of the product AB = α∈N n 0 c α ∂ α is a polynomial in a α and b α with |α| ≤ |γ|.The inverse B −1 = α∈N n (ii): In the proof of Theorem 2.15 (iii) we have already seen that the coefficients c γ of exp A = α∈N n 0 c α ∂ α depend polynomially on the coefficients a α of A = α∈N n 0 \{0} a α ∂ α with |α| ≤ |γ|.The condition d dt t=0 exp(tu) = u then follows by direct calculations.(iii): Follows like (ii) from Theorem 2.15 (iv).

Corollary 3 . 4 .
Let n ∈ N. If s ∈ S is represented by the signed measure μ and t ∈ S is represented by the signed representing measure ν then s * t is represented by the measure μ * ν.

Remark 3 . 6 .Corollary 3 . 7 .
For a general moment sequence s = (s α ) α∈N n 0 we only have s 0 > 0. But, scaling s = s −1 0 • s ∈ S gives s = exp(ln s 0 + log s).• The Fréchet topology on S and Corollary 3.4 imply the following.Let n ∈ N. The following hold: (i) The set S + is convex and closed.(ii) For all s, t ∈ S + we have s * t ∈ S + .Proof.(i): Convexity is clear.It suffices to prove that S + is closed in the Fréchet topology.Let s (n) = (s (n)

Definition 4 . 1 .
Let n ∈ N. We define the set D + := {A ∈ D | A is a positivity preserver} of all positivity preservers with constant coefficients and we define the set d + := {A ∈ d | exp(tA) ∈ D + for all t ≥ 0} of all generators of positivity preservers with constant coefficients.

Corollary 4 . 10 .
Let n ∈ N. Then d + log D + .Proof.We have log D + \ d + = ∅ by Example 4.9.We have seen in Main Theorem 4.7 the one-to-one correspondence between a positivity preserver A ∈ D + having an infinitely divisible representing measure and A ∈ D + having a generator.The infinitely divisible measures are fully characterized by the Lévy-Khinchin formula, see Theorem 2.18.The Lévi-Khinchin formula is used in the following result to fully characterize the generators d + of the positivity preservers D + .Main Theorem 4.11.Let n ∈ N. The following are equivalent:

Example 5 . 1 .
Let Δ = ∂ 2 x on L 2 ([0, ∞), R) with Dirichlet boundary conditions [27,] ≤d is a Fréchet space for all d ∈ N 0 since they are finite dimensional.Their Fréchet topology is unique.Example 2.4 (see e.g.[27, Ex.III]).Let n ∈ N. The vector space R[[x 1 , . . ., x n ]] of formal power series R to see that f is C 1 since log is smooth by Corollary 2.16 (iii).By Corollary 2.16 (ii) we have f = f .Hence, C 1 (D, d) coincides with the set of all C 1 -curves in D under the Fréchet topology of D.
4) and by Theorem 2.15 (ii) we also have that (D, • ) is a commutative group.By Corollary 2.16 (i) we have that (A, B) → AB −1 is continuous in the Fréchet topology.Hence, (D, • ) is an infinite dimensional commutative Fréchet Lie group.The properties about exp and log are Theorem 2.15 (iii)-(v).We now prove the regularity condition (vi) in Definition 2.5.Let F : R → D be a where f : R → d is the derivative Ḟ (t) of F (t) at t ∈ R, see [18, p. 10].But we can take the logarithm of F f (t) := log F (t) for all t ∈ ., A ∈ d + and therefore we have k ≤ 2.It is easy to see that the previous result also holds for n ≥ 2. To see this let α ∈ N n + is a closed and convex set.(ii)d+ is a non-trivial, closed, and convex cone.For the non-triviality we have the non-trivial Examples 4.2.For the convexity let A, B ∈ d + .Since (d, • , +) is a commutative algebra we have that exp(t(A + B)) = exp(tA) exp(tB) and since the product of two positivity preservers is again a positivity preserver we have A + B ∈ d + .For the closeness letA n ∈ d + for all n ∈ N 0 with A n → A in the Fréchet topology of R[[∂ 1 , ..., ∂ n ]],see Example 2.4.By Theorem 3.3 we have that exp : d → D is smooth, i.e., especially continuous.Hence,D + exp(tA n ) → exp(tA) ∈ D + for all t ≥ 0 since D + is closed by (i).Hence, A ∈ d + .For the cone property let A ∈ d + then also cA ∈ d + for all c ≥ 0.
0 with |α| ≥ 3. Then from Theorem 1.2 it follows that exp(∂ α ) ∈ D + .Choosing the same scaling argument (14) we find ∂ α ∈ d + .For D + and d + the following holds.Corollary 4.5.Let n ∈ N. Then the following hold:(i) D x, y) dy = (e t∂ 2 x p)(x) ∈ R[x] with T t x 2d+1 = p 2d+1 (x, t) for all d ∈ N 0 in [6, Dfn.3.1] and if additionally p ≥ 0 on [0, ∞), then T t p ≥ 0 on [0, ∞).On the other hand for even polynomials p ∈ R[x] we find that T t p∈C ∞ ([0, ∞)) \ R[x] but at least T t p ≥ 0 on [0, ∞).The reason for this strange behavior is of course the Dirichlet boundary condition and the resulting reflection principle.This effect needs further investigations.• This example shall also serve as a warning.Just because the operator (symbol) is ∂ 2 (20), ..., x n ] ≤|α|(20)thenAR[x 1 , ..., x n ] ≤d ⊆ R[x 1 , ..., x n ] ≤d for all d ∈ N 0 ,(21)i.e., A is called degree preserving.In fact for any linear A : R[x 1 , ..., x n ] → R[x 1 , ..., x n ] we have (20) ⇔ (21).(20)⇒(21) is clear.For (21) ⇒ (20) take in (20) the smallest α with respect to the lex-order such that deg q α > |α|.Then deg Ax α > |α| which is a contradiction to (21).Similar to Corollary 4.6 we have that the partial differential equation ∂ as in (20) is equivalent to a linear system of ordinary differential equations in the coefficients of p. Hence, it has a unique solution p( • , t) = exp(tA)p 0 for all t ∈ R, i.e., exp(tA) is well-defined for any A as in(20)and t ∈ R. If additionally A is a positivity preserver then exp(tA) is a degree preserving positivity preserver with polynomial coefficients for all t ≥ 0. In Main Theorem 4.11 we have already seen for the positivity preservers with constant coefficients that additional A appear.The restriction that A is degree preserving is a natural restriction since otherwise exp(tA)R[x 1 , . . ., x n ] ⊆ R[x 1 , . . ., x n ] for any t > 0. So we arrive at the following open problem.