Polya-Schur master theorems for circular domains and their boundaries

Theorem 1 below. The latter are linear transformations T on R[z] that are diag- onal in the standard monomial basis of R[z] and satisfy T (π(R)) ⊆π(R) ∪{0}. P´olya-Schur’s seminal paper generated a vast literature on this topic and related subjects at the interface between analysis, operator theory and algebra but a solu- tion to Problem 1 in the case Ω= R has so far remained elusive (cf. [12]). Among the most noticeable progress in this direction we should mention Theorem 17 of [28, Chap. IX],

Theorem 1 (P´olya-Schur theorem). Let λ : N →R be a sequence of real num- bers and T : R[z] →R[z] be the corresponding (diagonal) linear operator given by T (zn) = λ(n)zn, n ∈N. Define Φ(z) to be the formal power series Φ(z) = ∞ X k=0 λ(k) k! zk. The following assertions are equivalent: (i) λ is a multiplier sequence, (ii) Φ(z) defines an entire function which is the limit, uniformly on compact sets, of polynomials with only real zeros of the same sign, (iii) Either Φ(z) or Φ(−z) is an entire func

Theorem 1. Moreover, they also display an intimate connection between Problem 1 and its finite degree analog (Problem 2) in the case of (real) stability preservers. Notation 2. Given a linear operator T on C[z] we extend it to a linear operator – denoted again by T – on the space C[z, w] of polynomials in the variables z, w by setting T (zkwℓ) = T (zk)wℓfor all k, ℓ∈N. Definition 3. Let α1 ≤α2 ≤· · · ≤αn and β1 ≤β2 ≤· · · ≤βm be the zeros of two hyperbolic polynomials f, g ∈H1(R). We say that thes

Theorem 2. Let n ∈N and let T : Rn[z] →R[z] be a linear operator. Then T preserves hyperbolicity if and only if either (a) T has range of dimension at most two and is of the form T (f) = α(f)P + β(f)Q, f ∈Rn[z], where α, β : Rn[z] →R are linear functionals and P, Q ∈H1(R) have interlacing zeros, or (b) T [(z + w)n] ∈H2(R), or (c) T [(z −w)n] ∈H2(R). Real stable polynomials in two variables have recently been characterized by the authors [3] as the polynomials f(z, w) ∈R[z, w] that can be express

Theorem 3. Let n ∈N and let T : Cn[z] →C[z] be a linear operator. Then T : πn(R) →π(R) if and only if either (a) T has range of dimension at most one and is of the form T (f) = α(f)P, f ∈Cn[z], where α : Cn[z] →C is a linear functional and P ∈H1(R), or (b) T has range of dimension at most two and is of the form T (f) = ηα(f)P + ηβ(f)Q, f ∈Cn[z], where η ∈C, α, β : Cn[z] →C are linear functionals such that α(Rn[z]) ⊆ R, β(Rn[z]) ⊆R, and P, Q ∈H1(R) have interlacing zeros, or (c) There exists η ∈C

Theorem 4. Let n ∈N and let T : Cn[z] →C[z] be a linear operator. Then T preserves stability if and only if either (a) T has range of dimension at most one and is of the form T (f) = α(f)P, f ∈Cn[z], where α : Cn[z] →C is a linear functional and P ∈H1(C), or (b) T [(z + w)n] ∈H2(C). From Theorems 2 and 4 we deduce the following algebraic characterizations of hyperbolicity and stability preservers, respectively.

Corollary 1 (Algebraic Characterization of Hyperbolicity Preservers). A linear operator T : R[z] →R[z] preserves hyperbolicity if and only if either (a) T has range of dimension at most two and is of the form T (f) = α(f)P + β(f)Q, f ∈R[z], where α, β : R[z] →R are linear functionals and P, Q ∈H1(R) have inter- lacing zeros, or (b) T [(z + w)n] ∈H2(R) ∪{0} for all n ∈N, or (c) T [(z −w)n] ∈H2(R) ∪{0} for all n ∈N.

Corollary 2 (Algebraic Characterization of Stability Preservers). A linear operator T : C[z] →C[z] preserves stability if and only if either (a) T has range of dimension at most one and is of the form T (f) = α(f)P, f ∈C[z], where α : C[z] →C is a linear functional and P ∈H1(C), or (b) T [(z + w)n] ∈H2(C) ∪{0} for all n ∈N. Notation 3. To any linear operator T : C[z] →C[z] we associate a formal power series in w with polynomial coefficients in z GT (z, w) = ∞ X n=0 (−1)nT (zn) n!

Theorem 5 (Transcendental Characterization of Hyperbolicity Preservers). A lin- ear operator T : R[z] →R[z] preserves hyperbolicity if and only if either (a) T has range of dimension at most two and is of the form T (f) = α(f)P + β(f)Q, f ∈R[z], where α, β : R[z] →R are linear functionals and P, Q ∈H1(R) have inter- lacing zeros, or (b) GT (z, w) ∈H2(R), or (c) GT (z, −w) ∈H2(R).

Theorem 6 (Transcendental Characterization of Stability Preservers). A linear operator T : C[z] →C[z] preserves stability if and only if either (a) T has range of dimension at most one and is of the form T (f) = α(f)P, f ∈C[z], where α : C[z] →C is a linear functional and P ∈H1(C), or (b) GT (z, w) ∈H2(C).

Theorem 7. Let n ∈N and T : Cn[z] →C[z] be a linear operator. Let further C be an open circular domain given by C = Φ−1(H), where Φ is a M¨obius transformation as in (2.2). Then T : πn(C′) \ πn−1(C′) →π(C′) ∪{0} if and only if either (a) T has range of dimension at most one and is of the form T (f) = α(f)P, f ∈Cn[z], where α : Cn[z] →C is a linear functional and P ∈π(C′), or

Theorem 8. Let n ∈N and T : Cn[z] →C[z] be a linear operator and let further C = Φ−1(H) be an unbounded open circular domain, where Φ is a M¨obius trans- formation as in (2.2). Then T : πn(∂C) \ πn−1(∂C) →π(∂C) ∪{0} if and only if either (a) T has range of dimension at most one and is of the form T (f) = α(f)P, f ∈Cn[z], where α : Cn[z] →C is a linear functional and P ∈π(∂C), or (b) T has range of dimension two and the linear operator given by S = φ−1 m T φn is a stability preserver as in (b) of

Corollary 3 (Algebraic Characterization: Closed Circular Domain Case). Let T : C[z] →C[z] be a linear operator and let C ⊂C be an open circular domain given by C = Φ−1(H), where Φ is a M¨obius transformation as in (2.2). Then T : π(C′) → π(C′) ∪{0} if and only if either (a) T has range of dimension at most one and is of the form T (f) = α(f)P, f ∈C[z], where α : C[z] →C is a linear functional and P ∈π(C′), or (b) For all n ∈N the polynomial T (az + b)(cw + d) + (aw + b)(cz + d)
n is C-stable

Corollary 4 (Algebraic Characterization: Circle and Line Case). Let T : C[z] → C[z] be a linear operator and C = Φ−1(H) be an unbounded open circular domain, where Φ is a M¨obius transformation as in (2.2). Then T : π(∂C) →π(∂C) ∪{0} if and only if either (a) T has range of dimension at most one and is of the form T (f) = α(f)P, f ∈C[z], where α : C[z] →C is a linear functional and P ∈π(∂C), or (b) T has range of dimension two and for all n ∈N the linear operator given by Sn = φ−1 m(n)T φn is a

Theorem 9 (Hermite-Biehler theorem). Let h := f + ig ∈C[z], where f, g ∈R[z]. Then h ∈H1(C) if and only if f, g ∈H1(R) and g ≪f. Moreover, h is strictly stable if and only if f and g are strictly hyperbolic polynomials with no common zeros and g ≪f. The next theorem is often attributed to Obreschkoff[32].

Theorem 10 (Obreschkofftheorem). Let f, g ∈R[z]. Then αf +βg ∈H1(R)∪{0} for all α, β ∈R if and only if either f ≪g, g ≪f, or f = g ≡0. Moreover, αf +βg is strictly hyperbolic for all α, β ∈R with α2 + β2 ̸= 0 if and only if f and g are strictly hyperbolic polynomials with no common zeros and either f ≪g or g ≪f.

Lemma 1. Let n ∈N. Suppose that T : Rn+1[z] →R[z] preserves hyperbolicity and that f ∈R[z] is a strictly hyperbolic polynomial of degree n or n + 1 for which T (f) = 0. Then T (g) is hyperbolic for all g ∈R[z] with deg g ≤n + 1. Let T : Cn[z] →C[z] be a stability preserver and suppose that f ∈C[z] is a strictly stable polynomial of degree n for which T (f) = 0. Then T (g) is stable for all g ∈C[z] with deg g ≤n.

Lemma 2. Suppose that V ⊆R[z] is an R-linear space whose every non-zero element is hyperbolic. Then dim V ≤2. Suppose that V ⊆C[z] is a C-linear space whose every non-zero element is stable. Then dim V ≤1.

Lemma 3. Suppose that T : Rn[z] →R[z] maps all hyperbolic polynomials of degree at most n to hyperbolic polynomials. Then T is either stability preserving or stability reversing or the range of T has dimension at most two. In the latter case T is given by T (f) = α(f)P + β(f)Q, f ∈Rn[z], (3.1) where P, Q are hyperbolic polynomials whose zeros interlace and α, β are real-valued linear functionals on Rn[z].

Theorem 11 (Grace-Walsh-Szeg¨o coincidence theorem). Let f ∈C[z1, . . . , zn] be symmetric and multi-affine and let C be a circular domain containing the points ζ1, . . . , ζn. Suppose that either the total degree of f equals n or that C is convex (or both). Then there exists at least one point ζ ∈C such that f(ζ1, . . . , ζn) = f(ζ, . . . , ζ). From the Grace-Walsh-Szeg¨o coincidence theorem we immediately deduce:

Corollary 5. Let f ∈C[z1, . . . , zn] be of degree at most d in z1 and consider the expansion of f in powers of z1, f(z1, . . . , zn) = d X k=0 Qk(z2, . . . , zn)zk 1, Qk ∈C[z2, . . . , zn], 0 ≤k ≤d. Then f is stable if and only if the polynomial d X k=0 Qk(z2, . . . , zn)ek(x1, . . . , xd) d

Lemma 4. Let T : Cn[z1] →C[z1] be a linear operator such that T [(z1 + w)n] ∈ H2(C). If f ∈Hm(C) is of degree at most n in z1 then T (f) ∈Hm(C)∪{0}, where T is extended to a linear operator on C[z1, . . . , zm] by setting T (zα1 1 · · · zαm m ) = T (zα1 1 )zα2 2 · · · zαm m for all α ∈Nm with α1 ≤n (compare with Notation 2).

Corollary 1 then by Theorem 2 we have that T preserves hyperbolicity up to any degree n ∈N. Conversely, if T : R[z] →R[z] preserves hyperbolicity then for any n ∈N the restriction T : Rn[z] →R[z] preservers hyperbolicity (up to degree n). The case when dimR T (R[z]) ≤2 is clear. Suppose now that dimR T (R[z]) > 2 and that T [(z + w)n] ∈H2(R) for some n ∈N. Then by Lemma 4 we have that T [(z + w)m] ∈H2(C) ∪{0} for all m ≤n and since the latter polynomial has real coefficients we get T [(z + w)m] ∈H

Lemma 5 (Sz´asz). Let m, n ∈N with m ≤n and f(z) = Pn k=m ckzk ∈C[z]. If f(z) ∈H1(C) and cmcn ̸= 0 then for any r ≥0 one has |f(z)| ≤|cm|rm exp  r|cm+1| |cm| + 3r2 |cm+1|2 |cm|2 + 3r2 |cm+2| |cm|  whenever |z| ≤r. For k, n ∈N let (n)k = k! n

Theorem 12. Let F(z, w) = P∞ k=0 Pk(z)wk be a formal power series in w with polynomial coefficients. Then F(z, w) ∈H2(C) if and only if Pn k=0(n)kPk(z)wk ∈ H2(C) ∪{0} for all n ∈N.

Lemma 6. Let T : Cn[z] →Cm[z] be a linear operator and suppose m is minimal, i.e., m = max{deg T (f) : f ∈Cn[z]}. Let further C = Φ−1(H) be an open circular domain, where Φ is a M¨obius transformation as in (2.2), and let S : Cn[z] →Cm[z] be the linear operator defined by S = φ−1 m T φn. The following are equivalent: (i) T (f) is C-stable or zero whenever f is of degree n and C-stable, (ii) S(f) is H-stable or zero whenever f is of degree n and H-stable, (iii) S(f) is H-stable or zero whenever f

Lemma 7. Let f(z, w) ∈C[z, w] be of degree at most m in z and at most n in w and let Φ : C →H be a M¨obius transformation as in (2.2). If either (a) C is not the exterior of a disk, or (b) C is the exterior of a disk and (b1) the degree in z of φm,z(f)(z, w) is m, and (b2) the degree in w of φn,wφm,z(f)(z, w) is n, then f is H-stable if and only if φn,wφm,z(f) is C-stable.

Lemma 8. Let Ω1 be a path-connected subset of C and let Ω2 be a bounded subset of C. If T : Cn[z] →C[z] is a linear operator such that T : πn(Ω1)\πn−1(Ω1) →π(Ω2) then all polynomials in the image of πn(Ω1) \ πn−1(Ω1) have the same degree. Note that in the hypothesis of Lemma 8 we do not allow the identically zero polynomial to be in the image of πn(Ω1) \ πn−1(Ω1).

Corollary 3 and Corollary 4 are immediate consequences of Theorem 7 and The- orem 8, respectively. 4. Open Problems As we already noted in §1, Problems 1 and 2 have a long and distinguished history. In this paper we completely solved them for a particularly relevant type of sets, namely all closed circular domains and their boundaries. Among the most interesting remaining cases that are currently under investigation we mention: (a) Ωis an open circular domain, (b) Ωis a sector or a double sector