Catalan Generating Functions for Generators of Uni-parametric Families of Operators

In this paper we study solutions of the quadratic equation AY2-Y+I=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AY^2-Y+I=0$$\end{document} where A is the generator of a one parameter family of operator (C0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_0$$\end{document}-semigroup or cosine functions) on a Banach space X with growth bound w0≤14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_0 \le \frac{1}{4}$$\end{document}. In the case of C0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_0$$\end{document}-semigroups, we show that a solution, which we call Catalan generating function of A, C(A), is given by the following Bochner integral, C(A)x:=∫0∞c(t)T(t)xdt,x∈X,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C(A)x := \int _{0}^\infty c(t) T(t)x \; \mathrm{d}t, \quad x\in X, \end{aligned}$$\end{document}where c is the Catalan kernel, c(t):=12π∫14∞e-λt4λ-1λdλ,t>0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} c(t) := \frac{1}{2\pi } \int _{\frac{1}{4}}^\infty e^{-\lambda t} \frac{\sqrt{4\lambda -1}}{\lambda } \; \mathrm{d}\lambda , \quad t>0. \end{aligned}$$\end{document}Similar (and more complicated) results hold for cosine functions. We study algebraic properties of the Catalan kernel c as an element in Banach algebras Lω1(R+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1_{\omega }(\mathbb R^+)$$\end{document}, endowed with the usual convolution product, ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document} and with the cosine convolution product, ∗c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*_c$$\end{document}. The Hille–Phillips functional calculus allows to transfer these properties to C0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_0$$\end{document}-semigroups and cosine functions. In particular, we obtain a spectral mapping theorem for C(A). Finally, we present some examples, applications and conjectures to illustrate our results.


Introduction
The Catalan numbers (C n ) n≥0 given by, form an integer sequence deeply studied in number theory and combinatorics. Historically, one of the first interpretation given to the Catalan number C n 238 Page 2 of 27 A. Mahillo and P. J. Miana MJOM was through the number of ways to triangulate a regular n + 2-sided polygon, known as Euler problem. Another example where this sequence appears is counting the ways of constructing binary trees. Specifically, C n represents the number of ways to construct a binary tree with n nodes. In fact, a large amount of applications and interpretations of (C n ) n≥0 , more than 200, may be found in [19]. The generating function of the Catalan numbers is given by which is one of the solution for y in the quadratic equation, In this context, one may wonder what happens when we replace the complex values y and z by operators or elements in a general Banach algebra. Recently, this point of view has been explored in [14] for bounded operators in a Banach space X, obtaining a way to solve the equation presented above.
To extend these results to a wider family of operators, mainly nonbounded operators, we consider generators of C 0 -semigroups and cosine operators. A family of bounded operators (T (t)) t≥0 on a Banach space X is called a strongly continuous semigroup (or is given by the orbit u(t) = T (t)x. In the case that A ∈ B(X), the set of linear and bounded operators on X, the C 0 -semigroup is expressed by the vector-valued exponential function, is given by the orbit v(t) = Cos(t)x. In the case that A ∈ B(X) the cosine function is expressed by the vector-valued hyperbolic cosine function, see more details in [1,Section 3.14].
In this work we show that is a solution of the equation, where A is the generator of a C 0 -semigroup (T (t)) t≥0 bounded by T (t) ≤ Me w0t , with w 0 ≤ 1 4 , see Theorem 4.2. The function c, called Catalan kernel, is defined by Similarly, when A generates a cosine function (Cos(t)) t≥0 , Cos(t) ≤ Me w0t , with M ≥ 1 and w 0 ≤ 1 4 , the operator is a solution of the biquadratic equation The Catalan kernel c has already appeared in the literature and its integral expression (1.2), see for example [16, formula 15]. The key point is that the moments of this function are the Catalan numbers, Other integral representations of Catalan numbers may be found in [17,19]. However, other notable properties of this function have not yet been considered. In Sect. 2, the following algebraic properties are shown, for t > 0, see Theorem 2.4, Lemma 2.7 and Theorem 2.8. Here we denote by * and * c the usual convolution product and the cosine convolution product defined in the weighted Lebesgue space L 1 ω (R + ), see Sects. 2.1, and 2.2. The main idea in this paper is to obtain new information about the Catalan kernel c in the algebras (L 1 ω (R + ), * ) and (L 1 ω (R + ), * c ) (Sect. 2) to transfer later to C 0 -semigroups (Sect. 3) and cosine functions (Sect. 4).
The Laplace transform and the cosine transform are useful tools to obtain those properties for the Catalan kernel c. Also, spectra of the element c are identified and represented in both convolution algebras in Sect. 2.
For C 0 -semigroups, we define the Catalan operator in Sect. 3. We show a spectral mapping theorem for this operator and the connection with the square root in Theorem 3.2. In the case that A generates a C 0 -group, then 4A 2 generates a C 0 -semigroup and For cosine operators, we define the Catalan operator in Sect. 4. We also show a spectral mapping theorem for this operator and the connection with the square root in Theorem 4.4. As A also generates a C 0 -semigroup, C(4A) = (C(A)) 2 where C(4A) is given in Definition 3.1.
Finally, in the last section we present some concrete examples of operators A which generates C 0 -semigroups and cosine functions. We calculate the Catalan operator C(A) for these operators. We also give some conjectures and ideas to extend our results presented in a future research. For α-times integrated semigroup, resolvent estimates or fractional powers of infinitesimal generators of bounded C 0 -semigroups, the Catalan operator C(A) may be interesting to consider in further research.

Algebraic Properties of the Catalan Kernel
The Catalan numbers (C n ) n≥0 form a sequence of integers defined by the recurrence relation and C 0 = 1. They can also be expressed by the following explicit formula using binomial numbers, (2.1) and it satisfies the following quadratic equation, also satisfies this equation, Lastly, it's worth mentioning that the Catalan numbers admit the following integral representation, see [16,Equation 10].
Remind that a measurable function f belongs to this weighted Lebesgue space L 1 ω (R + ) if the following norm, is finite where ω ∈ R. In fact, the space L 1 ω (R + ) may be embedded with different convolution products.
The convolution product * is commutative, associative, with bounded approximate identity and . Then the space (L 1 ω (R + ), * ) is, in fact, a Banach algebra whose spectrum σ(L 1 ω (R + ), * ) = {z ∈ C | z ≥ −ω} and its Gelfand transform is the Laplace transform given by for f ∈ L 1 ω (R + ). As it is known, the Laplace transform verifies L(f * g) = L(f )L(g) and In the following theorem, we recollect some basic results about the Catalan kernel c. We also present an alternate definition for c using the complementary error function erfc defined by Since the following asymptotic behavior holds ([15, formula 40:6:3]), the erfc function belongs to L 1 ω (R + ) for ω < 1 4 , see its graphic in fig. 1. (2.4). Then the following properties hold. (iv) An alternative expression of c is given by

Theorem 2.2. Let the Catalan kernel c be defined by
Proof. The item (i) is an exercise of elemental calculus. To prove (ii), as where β is the Euler beta function, and we have used [4, formula 3.121 (2)].
As the Catalan kernel is a positive function, the Laplace transform of the function c is also checked in (ii) and Finally, to check (iv), we split the Laplace transform of c in the following way: , and the Laplace transform is injective in L 1 (R + ), we conclude the desired equality in L 1 (R + ) and then in L 1 and we plot ∂ σ (L 1 ω (R + ), * ) (c) in Fig. 2 for several values of ω. Let the Catalan kernel Notice that the Laplace transform of the Catalan kernel c is in fact the generating function of the Catalan numbers evaluated at −z, Eq. 2.1, that is, Using that the generating function C(z) satisfies the quadratic Catalan equation (2.2), we have that L(c) is a solution of the quadratic equation which motivates the study of the function c * c, with property (c * c) = −c that will be useful in the next section.
(iv) It admits the following representation in terms of the Catalan kernel, for z ≥ − 1 4 . As the Laplace transform is injective, we conclude the equality. Note that item (v) is a consequence of item (iv) To show item (vi), we apply Theorem 2.2 (4), and get .41]), we integrate by parts twice to compute I(t), i.e., By item (4), we conclude the following equality for t > 0.
Remark 2.5. Note that the function c * c is a positive, decreasing function and lim t→0 + (c * c)(t) = 1. We plot c * c in Fig. 3.

The Catalan Kernel in the Algebra
This product is also commutative, associative, with bounded approximate identity and Then the space (L 1 ω (R + ), * c ) is a Banach algebra whose Gelfand transform is the cosine transform given by The cosine transform is injective and verifies C(f * c g) = C(f )C(g) and see [11,Theorem 1.5].
In the next lemma we present two technical results about f * c2 and f * c3 , where f * c2 := f * c f and f * c3 := f * c2 * c f .
The first item is a direct consequence of the definition of cosine convolution product. To show (ii), note that As In the following lemma, we present some interesting algebraic properties about the Catalan kernel and the cosine convolution product * c .

Lemma 2.7. Let c be the Catalan kernel given in Definition
As an elemental exercise of calculus, we have that and we conclude the result. As (c * c c) (t) = 1 2 (c * c) + (c • c) , we apply (i) and Theorem 2.4(v) to have and we finish the proof of item (ii).
Finally, we present the main result of this section.

The Catalan Operator for C 0 -Semigroups
In this section we solve the general quadratic Eq. (1.1) in the case that A generates a C 0 -semigroup with growth bound less than 1 4 . To accomplish this we apply the Hille-Phillips functional calculus to the Catalan kernel c.
As σ(A) ⊂ {z ∈ C : z < ω}, a holomorphic function calculus (sometimes called Dunford-Schwartz calculus) is defined for holomorphic functions in a neighborhood of σ(A). This functional calculus is defined by the integral Cauchy-formula, As usual, the path Γ rounds the spectrum set σ(A). Both homomorphism, Θ(f ) and f (A) coincides under common conditions Θ(f ) = L(f )(−A), for "enough good functions" see for example [6].
In this section, we consider a C 0 -semigroup (T (t)) t≥0 with growth bound less than 1 4 . We start to give a formal definition for the Catalan operator for C 0 -semigroups.
where c is the Catalan kernel seen in Definition 2.1.
Recall, that if (T (t)) t≤0 is a uniformly bounded C 0 -semigroup with generator (A, D(A)) we have the following definition for the fractional power of the generator, see [20,section IX.11]. In the next theorem, we prove the main properties of the Catalan operator C(A) defined by C 0 -semigroups. (iii) The Catalan operator C(A) has the following integral representation (iv) The following representation holds, (v) The spectral mapping theorem holds for C(A), i.e., Proof. The proof of item (i) is a consequence of Theorem 2.2 (i). To show (ii), note that where we have applied Theorem 2.4 (v).
To show the item (iii), we have that 1 2π (iv) Now take x ∈ D(A). Then we have that Therefore, Finally, to show item (v), we write A ω = A − ω, and the function g ω defined by Note that C(A) = g ω (−A ω ) and the function Observe that we can extend holomorphically the function g ω to the set C\(−∞, ω − 1 4 ). In addition, note that g ω (0) = 1− which is well-defined for ω ≤ 1 4 and then g ω has finite polynomial limit at 0. Also, lim |z|→∞ g ω (z) = 0 with z ∈ C\(−∞, ω − 1 4 ) so g ω has polynomial limit at ∞. Using [6, Lemma 2.2.3] we have that the function g ω ∈ E ϕ0 , the extended Dunford-Riesz class, for ϕ 0 ∈ [ π 2 , π). Note that −A ω is a sectorial operator of angle π 2 because (e −ωt T (t)) t≥0 is a uniformly bounded C 0 -semigroup. Therefore, we can apply the spectral mapping theorem [6, Theorem 2.7.8] to obtain, and we conclude the proof.
In the case that A ∈ B(X) with 4A of power-bounded, we check that the definition of C(A) given in Definition 3.1 coincides with the power series presented in [14,Section 5]. Proof. Note that A generates a C 0 -semigroup, T (t) =: e tA and for t ≥ 0. Then where we have applied formula (1.4).
Remark 3.4. An alternative approach to Catalan operator C(A) may be followed using fractional powers of sectorial operators. As it is commented in the proof of Theorem 3.2 (v), −(A − ω) is a sectorial operator of angle π/2. Then B := I − 4A is also a sectorial operator of angle (at most) π/2, the fractional power √ B is sectorial of angle π/4, and its square is B. Using standard properties of fractional powers (see, for example [10]) one may establish that However this approach hides the rich algebraic properties of function c which are commented in Sect. 2. Even in the case that dim(X) = 2, the quadratic equation (1.1) may have one, two, infinite or no solutions, see [14,Section 6.1]. In the case that (AC(A)) −1 ∈ B(X), then this operator provides a second solution of (1.1). Note that which might give a way to apply the natural functional calculus treated in [6].
In the case that A and −A generates C 0 -semigroups, (T + (t)) t≥0 and (T − (t)) t≥0 it is said that A generates a C 0 -group (T (t)) t∈R given by Note that an algebra homomorphism Φ, which extends the map Θ, is defined by When A generates a bounded C 0 -group, (T (t)) t∈R , then A 2 generates a bounded C 0 -semigroup (T A 2 (t)) t>0 given by Theorem 3.5. Let A be the generator of a bounded C 0 -group, (T (t)) t∈R . Then Proof. As the operator A 2 generates a C 0 -semigroup, (T A 2 (t)) t≥0 given by (3.2), then 4A 2 also generates a bounded C 0 -semigroup, (T 4A 2 (t)) t≥0 and T 4A 2 (t) = T A 2 (4t) for t ≥ 0. By Definition 3.1, we have

The Catalan Operator for Cosine Functions
In this section we consider the general quartic Eq. (1.3) in the case that A generates a cosine operator with growth bound less than 1 4 . We follow similar (and more complicated) ideas than in the case of C 0 -semigroups.
We give formal definition for the Catalan operator for generators of cosine functions.
Finally, to show the item (iii), we have that 1 2π where we have applied formula (1.4). Now we suppose that A generates a C 0 -group (T (t)) t∈R such that T (t) ≤ Me ω |t| for t ∈ R and M ≥ 1 and ω ≥ 0. Then A 2 generates a cosine function (C(t)) t≥0 where Suppose that A is the generator of a cosine function (Cos(t)) t≥0 such that Cos(t) ≤ Me ωt for t ≥ 0, M ≥ 1 and ω > 0. Then A is the generator of a C 0 -semigroup (T (t)) t>0 where with T (t) ≤ Me ω 2 t for t > 0, see [1, Theorem 3.14.17]. Finally, we suppose that A ∈ B(X). As C(4A) and C(A) are bounded operators, σ(C(A)) = σ(C(4A)). We apply Theorem 3. Similarly, the equality holds for A ∈ B(X).