Catalan generating functions for bounded operators

In this paper, we study the solution of the quadratic equation TY2-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}$$TY^2-Y+I=0$$\end{document} where T is a linear and bounded operator on a Banach space X. We describe the spectrum set and the resolvent operator of Y in terms of the ones of T. In the case that 4T is a power-bounded operator, we show that a solution (named Catalan generating function) of the above equation is given by the Taylor series C(T):=∑n=0∞CnTn,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C(T):=\sum _{n=0}^\infty C_nT^n, \end{aligned}$$\end{document}where the sequence (Cn)n≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(C_n)_{n\ge 0}$$\end{document} is the well-known Catalan numbers sequence. We express C(T) by means of an integral representation which involves the resolvent operator (λT)-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\lambda T)^{-1}$$\end{document}. Some particular examples to illustrate our results are given, in particular an iterative method defined for square matrices T which involves Catalan numbers.


Introduction
The well-known Catalan numbers (C n ) n≥0 given by the formula appear in a wide range of problems.For instance, the Catalan number C n counts the number of ways to triangulate a regular polygon with n + 2 sides; or, the number of ways that 2n people seat around a circular table are simultaneously shaking hands with another person at the table in such a way that none of the arms cross each other, see for example [21,22].They have been studied in depth in many papers and monographs (see for example [2,16,20,22]) and the Catalan sequence is probably the most frequently encountered sequence.
The generating function of the Catalan sequence c = (C n ) n≥0 is defined by (1.1) This function satisfies the quadratic equation zy 2 − y + 1 = 0.The main object of this paper is to consider this quadratic equation in the set of linear and bounded operators, B(X) on a Banach space X, i.e., (1.2) where I is the identity on the Banach space, and T, Y ∈ B(X).Formally, some solutions of this vector-valued quadratic equations are expressed by which involves the (non-trivial) problems of the square root of operator 1 − 4T and the inverse of operator T .
In general, the equation (1.2) may have no solution, one, several or infinite solutions, see examples in Section 6.Note that study of quadratic equations in Banach space X which dim(X) ≥ 2 is much complicated than in the scalar case.For example there are infinite symmetric square roots of I 2 ∈ R 2×2 given by As far as we are aware, no useful necessary and sufficient conditions for the existence of solution of quadratic equations in Banach spaces are known, even in the classical case of square roots in finite-dimensional spaces.To find some easily applicable conditions is of interest, in part because these equations are frequently used in the study of, for example, physical or biological phenomena.
In 1952, Newton's method was generalized to Banach space by Kantorovich.Kantorovich's theorem asserts that the iterative method of Newton, applied to a most general system of nonlinear equations P (x) = 0, converges to a solution x * near some given point x 0 , provided the Jacobian of the system satisfies a Lipschitz condition near x 0 and its inverse at x 0 satisfies certain boundedness conditions.The theorem also gives computable error bounds for the iterates.From here, a large theory has been developed to obtain sharp iterative methods to approximate solutions of non linear equation (see for example [9,12,15]) and in particular quadratic matrix equations ( [8,11]).
The paper is organized as follows.In the second section, we show new results about the well-known Catalan numbers sequence (C n ) n≥0 .In Theorem 2.4, we prove that following technical identity holds for j ≥ 1.A nice result about solutions of quadratic equations are given in Theorem 2.1: the arithmetic mean y + z 2 is solution of the biquadratic equation 4x 2 w 4 − w 2 + 1 = 0. We consider the sequence c = (C n ) n≥0 as an element in the Banach algebra ℓ 1 (N 0 , 1  4 n ) in the third section.We describe the spectrum set σ(c) in Proposition 3.2 and the resolvent element (λ − c) −1 in Theorem 3.4.
In the forth section, we study spectral properties of the solution of quadratic equation (1.2) with T ∈ B(X).We prove several results between σ(T ) and σ(Y ) where σ(•) denotes the spectrum set of the operator T .Moreover, we express (λ − Y ) −1 in terms of the resolvent of operator T in Theorem 4.4.
For operators T which 4T are power-bounded, we define the generating Catalan function This operator solves the quadratic equation (1.2) and has interesting properties connected with T , see Theorem 5.1; in particular the following integral representation holds, In the last section we illustrate our results with some examples of operators T in the equation (1.2).We consider the Euclidean space C 2 and matrices We solve the equation (1.2) and calculate C(T ) for these matrices.We also check C(a) for some particular values of a ∈ ℓ 1 (N 0 , 1 4 n ) .Finally we present an iterative method for matrices R n×n which are defined using Catalan numbers.

Some news results about Catalan numbers
The Catalan numbers may be defined recursively by C 0 = 1 and (2.1) and first terms in this sequence are 1, 1, 2, 5, 14, 42, 132, . . . .The generating function of the Catalan sequence c = (C n ) n≥0 is given in (1.1).This function satisfies the quadratic equation see for example [22,Section 1.3].The second solution of this quadratic equation is given by The following theorem shows that the arithmetic mean of two solutions of these quadratic equations is also solution of a biquadratic equation, closer to the previous ones.
Theorem 2.1.Let A be a commutative algebra over R or C with x ∈ A. If y and z are solutions of the quadratic equations Proof.Note that it is enough to show that x 2 (y + z) 4 − (y + z) 2 + 4 = 0. We write for a while Since y and z are solutions of these quadratic equations, we have that xz 2 − xy 2 = 2 − (y + z) and and we may obtain y − z 2 in terms of y + z 2 whatever the inverse of 2x y + z 2 exists in the algebra A, i.e.
As a direct aplication of Abel's theorem to (1.1), we obtain that ( [22,Exercise A.66]).In fact one has that A straightforward consequence of the generating formula (1.1) and Theorem 2.1 is the following proposition, where we consider the odd and even parts, C o (z), and C e (z) of function C(z).The proof is left to the reader.
Catalan numbers have several integral representations, for example where the function β is the well-known Euler Beta function, β(u, v) := for u, v > 0, see the monography [22] and the survey [17].In the next theorem, we present a new results which involves the Taylor polynomials of the Catalan generating function C(z).
for j ≥ 1 and where the last equality holds for ℜ(z) ≥ 1 2 .
Proof.The first integral is a easy exercise of elemental calculus.To do the second one, note that and then for j ≥ 1.We iterate this formula to get the final expression.□ Remark 2.5.By holomorphic property, Theorem 2.4 holds for for j ≥ 1.Finally, when j → ∞, we recover the generating formula

The sequence of Catalan numbers
We may interpret the equality (2.3) in terms of norm in the weight Banach algebra ℓ 1 (N 0 , 1 4 n ).This algebra is formed by sequence a = (a n ) n≥0 such that and the product is the usual convolution * defined by The canonical base {δ j } j≥0 is formed by sequences such that (δ j ) n = δ j,n is the known delta Kronecker.Note that δ * n 1 = δ 1 . . .n δ 1 = δ n for n ∈ N.This Banach algebra has identity element, δ 0 , its spectrum set is the closed disc D(0, 1  4 ) and its Gelfand transform is given by the Z-transform It is straightforward to check that Z(δ n )(z) = z n for n ≥ 0 (see, for example, [13]).In the next proposition, we collect some properties of the Catalan sequence c in the language of the Banach algebra ℓ 1 (N 0 , 1 4 n ).In particular the identity (2.1) is equivalent to the item (iii).
Given λ ∈ C, we consider the geometric progression and Proof.By Proposition 3.1 (iii), (δ 0 − δ 1 * c) * c = δ 0 and we conclude that For λ ∈ Ω, we apply the Zeta transform to get that 4 ).To conclude the equality, we check that where we have applied the quadratic identity (2.In this section we study spectral properties of the solution of quadratic equation (1.2) with T ∈ B(X).We say that Y ∈ B(X) is a solution of (1.2) when the equality holds.Depend on T , the equation (1.2), has no solution, one, two or infinite solution, see subsection 6.1.
The proof of the following lemma is a direct consequence of the equality (1.2).
where we have applied (iv) and the equation (1.2). □ In the case that dim(X) < ∞, to be left-invertible implies to be invertible and the conditions of Theorem 4.2 hold.Corollary 4.3.Let X be a Banach space with dim(X) < ∞, T ∈ B(X) and Y a solution of (1.2).Then Y is invertible, T and Y commute and In the next theorem we give the expression of (λ − Y ) −1 which extend the equality Y To conclude the equality, we check that where we have applied the quadratic equation (1.2). (ii where we have applied the quadratic equation (1.2) in the last equality.□ Remark 4.5.The part (i) of Theorem 4.4 may be considered as an inverse spectral mapping theorem:

Catalan generating functions for bounded operators
In this section, we consider the particular case that T is a linear and bounded operator on the Banach space X, T ∈ B(X), such that i.e., 4T is a power-bounded operator.In this case σ(T ) ⊂ D(0, 1  4 ).Under the condition (5.1), we may define the following bounded operator, (5.2) as a direct consequence of (2.3).Moreover, the bounded operator C(T ) may be consider as the image of the Catalan sequence c = (C n ) n≥0 in the algebra homomorphism Φ : i.e., Φ(c) = C(T ).The Φ algebra homomorphism (also called functional calculus) has been considered in several papers, two of them are [5, Section 2] and more recently [6, Section 5.2].In particular, the map Φ allows to define the following operators where we have applied the "generalized binomial formula", (1 for α > 0. Remind that α n ∼ 1 n 1+α when n → ∞.Theorem 5.1.Given T ∈ B(X) such that 4T is power-bounded and c = (C n ) n≥0 the Catalan sequence.Then (iii) The following integral representation holds (iv) The spectral mapping theorem holds for C(T ), i.e, σ(C(T )) = C(σ(T )) and )) ⊂ σ(c).
Proof.(i) From (2.3), C(T ) ∈ B(X) as we have commented.It is clear that T and C(T ) commute.We apply the algebra homomorphism to the equality given in Proposition 3.1 (iii) to get (ii) As the homomorphism Φ is continuous, we apply the formula (3.1) to get n + 1 for n ≥ 0.

Examples, applications and final comments
In this section we present some particular examples of operators T for which we solve the equation (1.2).In the subsection 6.1, we consider the Euclidean space C 2 and matrices T = λI 2 , λ 0 1 1 0 , λ 0 1 0 0 where λ ∈ C. Note that we have to solve a system of four quadratic equations.We also calculate C(T ) for these matrices.In subsection 6.2 we check C(a) for some a ∈ ℓ 1 (N 0 , 1 4 n ).Finally we present an iterative method for matrices R n×n which are defined using Catalan numbers in subsection 6.3.
Now we study the case T = 0 λ λ 0 with λ ∈ C\{0}.The solutions of (1.2) are given by a where a is a solution of the biquadratic equation 4λ 2 a 4 − a 2 + 1 = 0.In the case that |λ| ≤ 1  4 , we get that where functions C e and C o are defined in Proposition 2.3 Finally take now T = 0 λ 0 0 with λ ∈ C. The only solution of (1.2) is given by Y 6.2.Catalan operators on ℓ p .We consider the space of sequences ℓ p (N 0 , 1 4 n ) where )), i.e., it is independent on p and coincides with the spectrum of the Catalan sequence c in ℓ 1 (N 0 , 1 4 n ) (Proposition 3.2).Now we consider the spaces ℓ p (Z) for 1 ≤ p ≤ ∞ defined in the usual way.The element a = δ 1 − δ 0 defines the classical backward difference operator for n ∈ Z.Note that ∥a∥ = 2, and see [7,Theorem 3.3 (4)].Now we need to consider a 8 and the associated Catalan generating operator defined by (5.2).By Theorem 5.1 (2), we get that where we have applied Theorem 2.4 for z = 3 2 .A similar results holds for the forward difference operator defined by ∆f (n) := f (n + 1) − f (n), [7, Theorem 3.2].6.3.Iterative methods on R n applied to Quasi-birth-death processes.The quadratic matrix equation: is related to the particular Markov chain characterized by its transition matrix P which is an infinite block tridiagonal matrix of the form: , where the blocks D 1 , D 2 , I, T are n × n nonnegative matrices such that D 1 +D 2 and I+T are row stochastic.A discrete-time Markov chain represent a quasi-birth-death stochastic process.In fact a quasi-birth-death stochastic process is a Discrete-time Markov chain having infinitely states ( [14]).Thus, a nonnegative solution of quadratic matrix equation (6.1) is necessary to describe probabilistically the behavior of that Markov chain.In [3,4] the author demonstrated the usefulness of Newton's method for solving the quadratic matrix equation.There are many papers containing algorithmic methodologies and acceleration techniques related to quadratic matrix equations, see for instance [3,4,9,11,18].
Our purpose in this section is to show experimentally the benefits of a higher order iterative method to approximate the nonnegative solution of equation (6.1) which uses the Catalan numbers, C j : Notice that method (6.2) has infinite speed of convergence to approximate a solution of equation (6.1), see [8].That is, the solution is obtained in the first iteration.To apply this method carries on computing the square root of the matrix (I − 4T ).To avoid this, we can truncate the series, thus obtaining a high-order method of convergence.This method can be write in terms of Sylvester equations, one of the most often in matrix equations ( [11]): is a bilinear constant operator.Notice that method (6.2) can be written in terms of generalized Sylvester equations, for j ≥ 1, (6.T (H 0 H j−1 + H j−1 H 0 ).
Notice that, method (6.7) is reduced to solve three Sylvester equations with the same matrix system.The Bartels-Stewart algorithm is ideally suited to the sequential solution of Sylvester equation (6.3) with the same matrix system ( [1]).
In particular, method of (6.2) truncated to k = 2 has forth order of converge: (6.Taking into account that the most commonly used Newton's method: only achieve quadratic convergence speed ( [3,4]), the forth order method (6.7) is a good alternative to approximate a solution of quadratic equation (6.1).Next, a numerical example is shown where the matrix T is ill conditioned.With high accuracy we approximate numerically the nonnegative solution of equation (6.1) using the method (6.7).To do that, we take T = (t ii ) a diagonal matrix 100 × 100 with entries t ii = 10 −1 , i = 1 . . ., 9 and t 10,10 = 10 −10 .Method (6.7) is implemented in Mathematica Version 10.0, with stopping criterion RES < 10 −20 , RES := ∥ Q(Y k )∥ ∞ .We choose the starting matrix Y 0 = T.We show the number of iterations necessary to achieve the required precision.The numerical results are reported in Table 1.

Lemma 4 . 1 .
Given T ∈ B(X) and Y a solution of (1.2).Then Y has left-inverse and Y −1 l = I − T Y .

(iv) T Y 2 =
Y T Y.Proof.(i) As 0 ∈ ρ(Y ), we obtain item (ii) from (1.2).The expression of T in (ii) implies that T and Y commute.Now, if T and Y commute, then the equality T Y 2 − Y T Y = 0 holds.Finally, we show that item (iv) implies (i).By lemma 4.1, we have that I − T Y is a left-inverse of Y and it is enough to check if is a right-inverse