# Exploring the spectra of some classes of paired singular integral operators: the scalar and matrix cases.

1 Introduction

Spectral theory has many applications in several different scientific areas and the importance of its study is globally acknowledged. For instance, Trefethen [49] said that "Eigenvalues are useful for three main reasons. The algorithmic reason is that if a matrix or linear operator can be diagonalized, transforming the problem to a basis of eigenfunctions, the solution of various problems may be speeded up. [...]. They give an operator a personality." There are thousands of articles and books on the role of the spectrum in Operator Theory. Andrew and Green [3] said that the spectral theory of operators on a Hilbert space is a rich, beautiful, and important theory. Beautiful graphical representations of the spectrum of some operators can be found in several works, for instance, in [4].

The development of the spectral theory is stimulated by the need to solve problems emerging from several fields in Mathematics and Physics. At present time, some progress has been achieved (see, for instance, [3], [4], [10], [11], [29], [33], [35], and [36]) for classes of singular integral operators whose properties allow the use of particular strategies in the study of the spectral problem. However despite several major developments, there is still no general and explicit method for obtaining the spectrum of any arbitrarly given singular integral operator. Also, the existing algorithms allow, in general, to study the spectrum of certain classes of singular integral operators but they are not designed to be implemented on a computer.

Factorization theory has a long and interesting history with roots that lie in the work of Plemelj [46] and is closely related to the spectral theory. Both theories have wide applications in: the study of Riemann-Hilbert boundary value problems, the Fredholm theory of singular integral operators, the theory of linear and non-linear differential equations, linear transport theory, the theory of diffraction of acoustic and electromagnetic waves, the theory of scattering and of inverse scattering, some branches of probability theory, among others (see, for instance, [1], [2], [14], [21], [25], [42], and [47]). In particular, even for the scalar case, one of the most important problems in factorization theory is the computation of the partial indices of factorable matrix functions. In turn, this problem is closely related to the theories of Wiener-Hopf systems of equations and of characteristic systems of singular integral equations with Cauchy kernel (see, for instance, [39], [41], and [44]). Similar to the case of spectral theory, the existing algorithms within the factorization theory show, in general, that it is possible to obtain some kind of factorization but are not designed to be implemented on a computer (see, for instance, [6], [7], [8], [9], [12], [13], [24], [26], [28], [34], [37], [38], [43], and [48]). Recently, we developed, and partially implemented on a computer, the generalized factorization algorithm [AFact] for special classes of essentially bounded matrix functions [20]. Due to its innovative character, the implementation of [AFact] potentiates the design of algorithms dedicated to specific domains of application. For instance, in [18] we have presented the [SInt] algorithm, a new calculation technique for computing some classes of Cauchy type singular integrals (important in the design of spectral algorithms). In [20] we presented the analytical algorithm [AEq] to solve integral equations concerning Hankel operators and directly related with the [AFact] algorithm. In [19] we described the explicit rational functions factorization algorithm [ARFact-Matrix] that computes explicit (left and right) factorizations of given non-singular rational matrix function defined on the unit circle. On the determination of the partial indices, some developments have also been made but, even in the rational case (and in recent publications), the methods are difficult to apply and were not designed to be implemented on a computer (see, for instance, [5], [12], [14], [27], [32], [34], [50], and [51]).

In addition, the vast majority of explicit analytical factorization methods depend on the knowledge of the zeros and poles of scalar functions. As a consequence, in many applications in real world, a numerical analysis of such methods is inevitable. However, due to many non-stability issues, such as the ones affecting the factorization partial indices (see, for instance, [19] and [28]), the numerical approach of Factorization Theory is a very difficult problem. Because of this fact, the design of new analytical methods, even if only for some restrict, special classes of matrix functions, is still very significant to the development of such theory. As an example, due to the symbolic and numeric capabilities of Mathematica, the [ARFact-Scalar] algorithm [19] always computes the factorization index of the considered non-singular scalar rational function.

In recent years, several software applications were made available to the general public with extensive capabilities of symbolic computation. These applications, known as computer algebra systems (CAS), allow to delegate to a computer all, or a significant part, of the symbolic calculations present in many mathematical algorithms. In our work we use the CAS Mathematica (2) to implement for the first time on a computer analytical algorithms developed by us and others authors within Operator Theory (see, for instance, [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [32], and [45]). In the last years we designed and/or implemented analytical algorithms for solving integral equations, analytical algorithms to factorize scalar and matrix functions, calculation techniques to compute singular integrals, and more recently analytical algorithms to study the spectrum and the kernel of several classes of singular integral operators. It is our belief that the construction and implementation on a computer of these kind of analytical algorithms is a very interesting line of research.

In this paper we present some results on the spectra of paired singular integral operators with essentially bounded matrix functions coefficients, defined on the unit circle. In addition, we describe analytical algorithms that allows us to study the spectra of some classes of paired singular integral operators, with rational coefficients, defined on the unit circle. It is shown how the symbolic computation capabilities of Mathematica can be used to check, for each considered class of singular integral operators with rational coefficients, if a complex number, chosen arbitrarily, belongs to its spectrum. For the scalar case the automated checking process is called [ASpecPaired-Scalar] algorithm (3). For the matrix case, we designed the analytical algorithm [ASpecPaired-Matrix] which in turn makes use of the [ARFact-Matrix] algorithm [19].

The remainder of this paper is organized as follows:

In Section 2 it is explained how the study of the factorability of scalar and matrix functions is related to the invertibility of certain classes of singular integral operators and, consequently, to the study of the spectra of singular integral operators. Some new results on the spectra of paired singular integral operators are given. In Section 3 we describe the spectral algorithm [ASpecPaired-Scalar], that explores the spectra of one-dimensional paired singular integral operators, with rational coefficients, defined on the unit circle, and we present some nontrivial examples computed with this algorithm.

Section 4 is dedicated to the formal description of the [ASpecPaired-Matrix] algorithm, that explores the spectra of paired singular integral operators, with rational matrix functions coefficients, defined on the unit circle. Since the source code of [ASpecPaired-Matrix] uses the analytical algorithm [ARFact-Matrix] (see [19]) to compute left and right factorizations of non-singular rational matrix functions, a brief description of this second algorithm is made in subsection 4.1. Then some nontrivial examples computed with the algorithm are given.

Section 5 is devoted to special classes of paired singular integral operators with essentially bounded coefficients, defined on the unit circle. We present new results that relate the spectra of these operators with the spectra of a special class of self-adjoint singular integral operators. We also explain how the generalized factorization algorithm [AFact] (see [20]), for special classes of essentially bounded matrix functions, can be used to explore the spectra of some particular classes of paired singular integral operators and give some nontrivial examples.

Section 6 contains some final remarks about our current work and related lines of research that we find potentially interesting.

2 Paired singular integral operators

Singular integral operators (4) are classic mathematical objects with a vast array of applications in a large range of scientific research areas (see, for instance, [30], [31], [37], [40], [41], and [47]). There exist several numerical algorithms and approximation methods for evaluating certain classes of singular integrals. Also, there are several analytical techniques that allow the exact computation of singular integrals for particular cases. However, the [SInt] and [SIntAFact] algorithms described in [18] are the only analytical algorithms, up to our knowledge, written and implemented for computing singular integrals with general functions (5). On the other hand, the study of the spectra of paired singular integral operators is supported by the Factorization Theory. Thus, we use some of our factorization algorithms (see, for instance, [19] and [20]) to design the spectral algorithms described in the present paper. In the remainder of this paper the concepts and results will be presented (almost entirely) in the matrix case. However, whenever appropriate, the scalar case will be analyzed in a specifically way.

Let T denote the unit circle in the complex plane. Let [T.sub.+] and [T.sub.-] denote the open unit disk and the exterior region of the unit circle ([infinity] included), respectively. It is well known that the singular integral operator with Cauchy kernel, [S.sub.T]t, defined almost everywhere on T by

[S.sub.T][phi](t) = 1/[pi]i [[integral].sub.T] [[phi]([tau])]/[[tau]-t]d[tau], t [member of] T, (2.1)

where the integral is understood in the sense of its principal value, represents a bounded linear operator in [L.sub.2](T). In addition, [S.sub.T] is a selfadjoint and unitary operator in the Lebesgue space [L.sub.2](T) (see, for instance, [31] and [37]). Thus, we can associate with this operator two complementary Cauchy projection operators

[P.sub.[+ or -]] = (I [+ or -] [S.sub.T])/2, (2.2)

where I represents the identity operator. Obviously, we have that

[S.sub.T] = 2[P.sub.+] - I, [P.sub.+] - [P.sub.-] = [S.sub.T], and [P.sub.+][P.sub.-] = [P.sub.-][P.sub.+] = 0. (2.3)

The projectors (2.2) allow us to decompose the space [L.sub.2](T) into the topological direct sum

[L.sub.2](T)= [L.sup.+.sub.2](T) [symmetry] [L.sup.-0.sub.2](T),

where

[L.sup.+.sub.2](T) = im[P.sub.+] and [L.sup.-0.sub.2](T) = im[P.sub._].

We also consider the space

[L.sup.-.sub.2](T) = [L.sup.-,0.sub.2](T) [symmetry] C.

Let [L.sub.[infinity]](T) be the space of all essentially bounded functions on the unit circle.

By [L.sup.+.sub.[infinity]](T) and [L.sup.-.sub.[infinity]](T) we denote the sets of functions which are holomorphic and bounded on [T.sub.+] and [T.sub.-], respectively. It follows that [L.sup.[+ or -].sub.[infinity]](T) [subset] [L.sup.[+ or -].sub.2](T).

Let R(T) be the algebra of rational functions without poles on T and let [R.sub.[+ or -]](T)

denote the subsets of R(T) whose elements are without poles in[T.sub.[+ or -]].

Let [phi], [psi] [member of] [[[L.sub.[infinity]](T)].sub.n,n]. Operators of the form T = [phi]I + [psi][S.sub.T] and T = [phi]I + [S.sub.T][phi]I are linear and bounded singular integral operators (see, for instance, [31]). In the following, these operators will be written in a more convenient form as

[T.sub.{a,b}] = a[P.sub.|] + b[P.sub._] (2.4)

and

[[??].sub.{a,b}] = [P.sub.+]aI + [P.sub._]bI, (2.5)

where a = [phi]+[phi] and b = [phi] - [phi]. We will call these operators, paired singular integral operators, with coefficients a and b.

2.1 Factorization of functions: scalar and matrix cases

Now let us introduce the concept of a generalized factorization for matrix functions (see, for instance, [14] and [42]): we say that a matrix function r [member of] [[[L.sub.[infinity]](T)].sub.n,n], that is, a matrix function whose entries are essentially bounded functions on the curve T, admits a left (right) generalized factorization in [L.sub.2]( T) if it can be represented as

r = [r.sub.+][LAMBDA][r.sub._] (r = [r.sub._][LAMBDA][r.sub.+]), (2.6)

where

[mathematical expression not reproducible]

[[chi].sub.j] [member of] Z, j = 1,n, with [[chi].sub.1] [greater than or equal to] [[chi].sub.2] [greater than or equal to] ... [greater than or equal to] [[chi].sub.n], and [r.sub.+][P.sub.+][r.sub._]I ([r.sub._][P.sub.+][r.sub.+]I) represents a bounded linear operator in [[[L.sub.2](T)].sub.n]; the number [chi] = [n.[summation] over j=1] [[chi].sub.j]is called the factorization index of the determinant of the matrix function r. The integers [[chi].sub.j] are called its left (right) partial indices. If [x.sub.j] = 0, j = [bar.1,n], then r is said to admit a left (right) canonical generalized factorization (can. gen. fact.).

Any non-singular rational matrix function r [member of] [[R(T)].sub.n,n] admits a left (right) generalized factorization of the form (2.6) (see, for instance, [28]), where

[r.sup.[+ or -]1.sub.+] [member of] [[[R.sub.+](T)].sub.n,n], [r.sup.[+ or -]1.sub.-] [member of] [[[R.sub.-](T)].sub.n,n].

For the particular rational scalar case we note that

[chi] = [z.sub.+] - [p.sub.+]; (2.7)

where [z.sub.+] is the number of zeros of r in [T.sub.+] (with regard to their multiplicities) and [p.sub.+] is the number of poles of r in [T.sub.+] (with regard to their multiplicities) (see, for instance, [19]).

Remark 2.1.

(i) A natural and nontrivial question arises concerning the relation of the left and right partial indices of a generalized factorization of a matrix function r. It is well known that the sum of the left partial indices and the sum of the right partial indices are equal, that is, the factorization index x is uniquely determined by a given matrix function r. It was proved in [27] that this relation is the only existing one between the sets of the left and right partial indices.

(ii) The left (right) partial indices [x.sub.i] are uniquely determined by the matrix function r, that is, in a factorization of the form (2.6), matrix A is uniquely defined. However, this is not true for the factors [r.sub.[+ or -]] although a general relation between the factors for distinct generalized factorizations of the same given matrix function r can be found, for instance, in [28].

2.2 On the invertibility of paired singular integral operators

In this subsection we will see how the study of the factorability of scalar and matrix functions is related to the invertibility of certain classes of singular integral operators and, consequently, to the study of its spectra.

The following results show the importance of calculating the partial indices of matrix functions for the study of invertibility of paired singular integral operators. For the scalar case, the calculation of the factorization index is always possible in the rational case, by using the [ARFact-Scalar] algorithm (see Subsection 3.1). We obtained the following result which relates directly operators [T.sub.{a,b}] and [[??].sub.{a,b}] defined in (2.4) and (2.5) to one another (for the scalar case see, for instance, [31]).

Theorem 2.2. Let a,[b.sup.[+ or -]1] [member of] [[[L.sub.[infinity]](T)].sub.n,n]. If ab = ba, then the operators [T.sub.{a,b}] and [[??].sub.{a,b}] are related by the equality

[Z.sub.1][T.sub.{a, b}][Z.sub.2] = [[??].sub.{a ,b}], (2.8)

where [Z.sub.1] and [Z.sub.2] are the invertible operators (6)

[Z.sub.1] = (I + [P.sub.+]a[b.sup.-1][P.sub._]) [b.sup.-1]I and [Z.sub.2] = (I - [P.sub._]a[b.sup.-1][P.sub.+]) bI.

Proof. It is easy to see that [Z.sup.-1.sub.1] = b (I - [P.sub.+]a[b.sup.-1] [P.sub.-]).

Using properties (2.3) we get that

[Z.sup.-1.sub.1][[??].sub.{a,b}] = b[P.sub.+]aI + b[P.sub._] bI - b[P.sub.+]a[b.sup.-1][P.sub._]bI.

Considering that [P.sub.+] = I - [P.sub.-], we can rewrite

[Z.sup.-1.sub.1][[??].sub.{a,b}] = baI - b[P.sub._]aI + b[P.sub._]bI - ba[b.sup.-1][P.sub._]bI + b[P.sub._]a[b.sup.-1][P.sub._]bI.

Since ab = ba, we get

[Z.sup.-1.sub.1] [[??].sub.{a,b}] = abI + b[P.sub.-]bI - a[P.sub.-]bI - b[P.sub.-]a[b.sup.-1][P.sub.+]bI

= a[P.sub.+]bI + b[P.sub.-]bI - b[P.sub.-]a[b.sup.-1][P.sub.+]bI

= [T.sub.{a,b}][Z.sub.2] [??]

Corollary 2.3. Let a, [b.sup.[+ or -]1] [member of] [[[L.sub.[infinity]](T)].sub.n,n]. If a and b are simultaneously diagonalizable matrix functions, then relation (2.8) holds.

Proof. If a and b are simultaneously diagonalizable matrix functions, then exists a nonsingular matrix function v such that both [v.sup.-1]av and [v.sup.-1] bv are diagonal matrix functions.

Let [d.sub.1] = [v.sup.-1]av and [d.sub.2] = [v.sup.-1]bv then it follows that

ab = v[d.sub.1][v.sup.-1]vd2[v.sup.-1] = v[d.sub.1][d.sub.2][v.sup.-1] = v[d.sub.2][d.sub.1][v.sup.-1] = v[v.sup.-1]bv[v.sup.-1]av[v.sup.-1] = ba [??]

Corollary 2.4. If a, [b.sup.[+ or -]1] [member of] [L.sub.[infinity]](T), then relation (2.8) holds.

Let e denote the n x n identity matrix function.

The next results on the invertibility of the singular integral operator [T.sub.{a,b}] are known (see, for instance, [43]).

Theorem 2.5. If [r.sup.[+ or -]1.sub.+] [member of] [[[L.sup.+.sub.[infinity]](T)].sub.n,n] and [r.sup.[+ or -]1.sub.-] [member of] [[[L.sup.-.sub.[infinity]](T)].sub.n,n], then [mathematical expression not reproducible] is an invertible operator with inverse

[mathematical expression not reproducible]

Theorem 2.6. Let [a.sup.[+ or -]1],[b.sup.[+ or -]1] [member of][[[L.sub.[infinity]](T)].sub.n,n] and r = [b.sup.-1]a. Moreover, let r = [r.sub.-][LAMBDA][r.sub.+] be a right generalized factorization of the matrix function r. Then a generalized inverse of the operator [T.sub.{a,b}] is given by

[mathematical expression not reproducible] (2.9)

In addition, the operator is invertible (left-sided invertible, right-sided invertible) if and only if all right partial indices of r are zero (non-negative, non-positive). The inverse (a left inverse, a right inverse) operator is given by (2.9).

Remark 2.7. A left generalized factorization of the matrix function [a.sup.-1]b can also be considered and the operator [mathematical expression not reproducible] is obtained as a generalized inverse of the operator [T.sub.{a,b}]. This operator is the inverse (a left inverse, a right inverse) operator if and only if all left partial indices of r are zero (non-positive, non-negative).

Based on the same ideas we formulated similar results for the singular integral operator [[??].sub.{a,b}] defined in (2.5).

Theorem 2.8. If [r.sup.[+ or -]1.sub.+] [member of] [[[L.sup.+.sub.[infinity]](T)].sub.n,n] and [r.sup.[+ or -]1.sub.-] [member of] [[[L.sup.-.sub.[infinity]](T)].sub.n,n], then [mathematical expression not reproducible] is an invertible operator with inverse

[mathematical expression not reproducible]

Theorem 2.9. Let [a.sup.[+ or -]1],[b.sup.[+ or -]1] [member of] [[[L.sub.[infinity]](T)].sub.n,n] and r = a[b.sup.-1]. Moreover, let r = [r.sup.-][LAMBDA][r.sub.+] be a right generalized factorization of the matrix function r. Then a generalized inverse of the operator [[??].sub.{a,b}] is given by

[mathematical expression not reproducible] (2.10)

In addition, the operator is invertible (left-sided invertible, right-sided invertible) if and only if all right partial indices of r are zero (non-negative, non-positive). The inverse (a left inverse, a right inverse) operator is given by (2.10).

Remark 2.10. A left generalized factorization of the matrix function b[a.sup.-1] can also be considered and the operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE] is obtained as a generalized inverse of the operator [[??].sub.{a,b}]. This operator is the inverse (a left inverse, a right inverse) operator if and only if all left partial indices of r are zero (non-positive, non-negative).

For the scalar case, Theorem 2.6 and Theorem 2.9 can be stated as follows (see, for instance, [31]).

Theorem 2.11. Let [a.sup.[+ or -]1],[b.sup.[+ or -]1] [member of] [L.sub.[infinity]](T) and r = a[b.sup.-1]. Moreover, let r = [r.sub._][t.sup.x]r+ be a generalized factorization of the function r. The operators [T.sub.{a,b}] and [[??].sub.{a,b}] are invertible, only left invertible or only right invertible depending on whether the number x is equal to zero, positive or negative, respectively. In any case, the inverse operators of the operators [T.sub.{a,b}] and [[??].sub.{a,b}], from the corresponding side are given by

[mathematical expression not reproducible] (2.12)

2.3 On the spectra of paired singular integral operators

In this subsection we will see how the study of the factorability of scalar and matrix functions is related to the study of the spectra of the paired singular integral operators [T.sub.{a,b}] and [[??].sub.{a,b}] defined in (2.4) and (2.5). The spectrum of a bounded linear operator T is a closed, bounded, and non-empty subset of C, defined by

[sigma](T) = [[lambda] [member of] C : T - [lambda]I is not a bounded invertible operator}

Using the same type of calculations used to prove Theorem 2.2 and Corollary 2.3 we proved the next results.

Theorem 2.12. Let a,b [member of] [[[L.sub.[infinity]](T)].sub.n,n]. If ab = ba, then

[sigma]([T.sub.{a,b}]) = [sigma]([[??].sub.{a,b}]). (2.13)

Corollary 2.13. Let a,b [member of] [[[L.sub.[infinity]](T)].sub.n,n].

(i) If a and b are matrix functions simultaneously diagonalizable, then the relation (2.13) hold.

(ii) If a and b are scalar functions, then relation (2.13) hold.

Remark 2.14. When ab = ba, the equality (2.13) is not necessarily satisfied (see Example 4.1.2).

It is obvious that, if a or b is a constant function, i.e., a(t) [equivalent to] c or b(t) [equivalent to] c, for c [member of]C, then c [member of][sigma]([T.sub.{a,b}]) and c [member of][sigma]([[??].sub.{a,b}]).

Using Theorem 2.6 and Remark 2.7, we formulate the following result on the spectra of [T.sub.{a,b}] and [[??].sub.{a,b}].

Theorem 2.15. Let a, b [member of] [[[L.sub.[infinity]](T)].sub.n,n].

(i) If det(a(t) - [[lambda].sub.1]e) [equivalent to]0 ([[lambda].sub.1] [member of]C) and det(b(t) - [[lambda].sub.1]e) [equivalent to] 0 ([[lambda].sub.1] [member of]C), then [sigma]([T.sub.{a,b}]) - [sigma]([[??].sub.{a,b}]) = {[[lambda].sub.1], [[lambda].sub.1]}. (2.14)

(ii) If det(a(t) - [lambda]e) [??] 0 ([for all][lambda] [member of]C), then

[sigma]([T.sub.{a,b}]) = {[lambda] [member of]C : [(a - [lambda]e).sup.-1](b - [lambda]e) does not admit a left can. gen. fact.}. (2.15)

[sigma]([[??].sub.{a,b}]) = {[lambda] [member of]C : [(b - [lambda]e).sup.-1][(a - [lambda]e).sup.-1] does not admit a left can. gen. fact.}. (2.16)

(iii) Ifdet(b(t) -[lambda]e) [??]0 ([for all][lambda] [member of] C), then

[sigma]([T.sub.{a,b}]) = {[lambda] [member of] C : [(b-[lambda]e).sup.-1](a-[lambda]e) does not admit a right can. gen. fact.}. (2.17)

[sigma]([[??].sub.{a,b}]) = {[lambda] [member of] C : (a-[lambda]e)[(b-[lambda]e).sup.-1] does not admit a right can. gen. fact.}. (2.18)

Let us see now some particular cases that illustrate (i) of Theorem 2.15. Cauchy type singular integral operator

It is well known that the spectrum of a self-adjoint operator is a nonempty subset of R. Also, the spectrum of a unitary operator lies always on the unit circle. Let [[phi].sub.+] [member of][[[L.sup.+.sub.[infinity]](T)].sub.n,n] and [[phi].sub.-] [member of][[[L.sup.-.sub.[infinity]](T)].sub.n,n] such that [[phi].sub.-]([infinity]) = 0. Since [S.sub.T][[phi].sub.+] = [[phi].sub.+] and [S.sub.T][[phi].sub.-] = -[[phi].sub.-], for the Cauchy type singular integral operator, we get

[sigma]([S.sub.T]) = {-1,1}.

Projection operators

Considering the relations between a projection operator P and its complementary operator Q = I - P,

(P -[lambda]I) ([(1 -[lambda]).sup.-1] P -[[lambda].sup.-1] Q) = ([(1-[lambda]).sup.-1]P -[[lambda].sup.-1]Q) (P - [lambda]I) = I

we get (7) that

[sigma](P) [subset]{0,1}.

* Identity operator: Considering the identity operator I, it is obvious that

[sigma](I)= {1}

* Cauchy projection operators: Considering the Cauchy projection operators [P.sub.[+ or -]] defined in (2.2), it is obvious that ker([P.sub.+]) = [L.sup.-.sub.2](T) and im([P.sub.+]) = [L.sup.+.sub.2](T) and, as a consequence, we get that

[sigma]([P.sub.[+ or -]]) = {0,1}

3 Spectral algorithm: the rational scalar case

This section is dedicated to the formal description of the [ASpecPaired-Scalar] algorithm, that explores the spectra of one-dimensional paired singular integral operators, with rational coefficients, defined on the unit circle.

3.1 [ASpecPaired-Scalar] algorithm

In this subsection it is shown how the symbolic and numeric computation capabilities of the computer algebra system Mathematica were used to construct a spectral algorithm to explore the spectra of one-dimensional paired singular integral operators of the form

[T.sub.{a,b}] = a[P.sub.+] + b[P.sub.-] and [[??].sub.{a,b}] = [P.sub.+]aI + [P.sub.-]bI,

defined in (2.4) and (2.5), respectively, with a, b [member of] R(T).

The [ASpecPaired-Scalar] algorithm checks if a complex number (chosen arbitrarily) belongs to the spectra of operators [T.sub.{a,b}] and [[??].sub.{a,b}]. The implementation of this spectral algorithm with the Mathematica software system allows to obtain in a simpler and more immediate way the results of lengthy and complicated calculations.

In the design of the spectral algorithm we used parts of the code of the analytical algorithm [ARFact-Scalar] (see [19]), that computes explicit factorizations for any factorable rational scalar function defined on the unit circle.

This spectral algorithm has a rather simple structure since the knowledge of the factorization index [kappa] of a non-singular scalar rational function, that can be determined by formula (2.7), is the only information the algorithm requires to determine if a complex number is in the spectra of operators [T.sub.{a,b}] and [[??].sub.{a,b}] (see Theorem 2.15). In addition, the symbolic computation capabilities of Mathematica, and the pretty-print functionality (8), allow the [ASpecPaired-Scalar] code to be very simple and syntactically similar to its analytical counterpart.

The [ASpecPaired-Scalar] algorithm can be applied to any given one-dimensional paired singular integral operator of classes [T.sub.{a,b}] and [[??].sub.{a,b}] defined in (2.4) and (2.5), with rational coefficients.

For each pair of inputed functions a, b [member of] R(T), and complex value A, chosen arbitrarly, the [ASpecPaired-Scalar] algorithm gives the output

[[lambda] [member of] [sigma]([T.sub.{a,b}]) and [lambda] [member of] [[??].sub.{a,b}]] or [[lambda] [??] [sigma]([T.sub.{a,b}]) and [lambda] [??] [[??].sub.{a,b}]].

The pseudo code of the [ASpecPaired-Scalar] algorithm is shown in Figure 2.

To check whether a given complex number belongs to the spectrum of the paired singular integral operators the algorithm works as follows.

The [ASpecPaired-Scalar] algorithm checks whether any of the functions a or b is the constant function with value A. If this happens, then the output is that the complex number A belongs to the spectra of operators [T.sub.{a,b}] and [[??].sub.{a,b}]. Otherwise, the algorithm constructs the auxiliary rational function r(t) = (a(t) - [lambda])[(b(t) - [lambda]).sup.-1]. (3.1)

If r is not a factorable function (with respect to the unit circle), that is, if at least one of its zeros or poles lies in T, then the output is [[lambda] [member of] [sigma]([T.sub.{[alpha],b}]) and [lambda] [member of] [sigma]([[??].sub.{[alpha],b}]]. If the rational function (3.1) admits a factorization of the form (2.6), then its factorization index x is computed through formula (2.7).

With the knowledge of x and using Theorem 2.15 the spectral algorithm can now conclude if the complex number A is in the spectra of operators [T.sub.{a,b}] and [[??].sub.{a,b}]. The algorithm gives the output [[lambda] [??] [sigma]([T.sub.{a,b}]) and [lambda] [??] [sigma]([[??].sub.{a,b}])] if x = 0. Else, the output is [[lambda] [member of] [sigma]([T.sub.{a,b}]) and [lambda] [member of] [sigma]([[??].sub.{a,b}])].

Therefore, the analysis of the code reveals that one key step in this algorithm is the computation of the zeros and poles (with regard to their multiplicities), of the rational function (3.1) and whether they lie in T, [T.sub.+], or [T.sub.-].

We note that, since the zeros and poles of r(t) are a crucial information for this spectral algorithm, the success of the [ASpecPaired-Scalar] algorithm depends on the possibility of finding those zeros and poles by solving polynomial equations. This can be a serious limitation when working with polynomials of fifth degree or higher. However, even in this case, thanks to the symbolic and numeric capabilities of Mathematica, it is still possible to check if a complex number A (chosen arbitrarily) belongs to the spectra of operators [T.sub.{a,b}] and [[??].sub.{a,b}].

Remark that Mathematica uses Root objects to represent solutions of algebraic equations in one variable, when it is impossible to find explicit formulas for these solutions.

The Root object is not a mere denoting symbol but rather an expression that can be symbolically manipulated and numerically evaluated with any desired precision. In particular, it is still possible to know if any given lies in T, [T.sub.+], or [T.sub.-] (see Figure 3). In pratical terms, this means that the factorization index of r (when it exists) is always obtained explicitly by the spectral algorithm, and this is all the information the [ASpecPaired-Scalar] algorithm requires to conclude if a given complex number is in the spectra of the operators.

3.2 [ASpecPaired-Scalar] examples

In this subsection we present nontrivial examples computed by the [ASpecPaired-Scalar] algorithm.

Example 3.1. Let us consider the paired singular integral operators [T.sub.{a,b}] and [[??].sub.{a,b}] defined in (2.4) and (2.5) with rational scalar coefficients

a(t) = 3[t.sup.3] - 5[t.sup.2] and b(t) = [t.sup.6] - 3[t.sup.4] + [t.sup.3] - 2[t.sup.2] + 2t + 3.

We want to check if the complex number A = 4 belongs to the spectra of those operators.

The [ASpecPaired-Scalar] algorithm constructs the auxiliary function

r(t) =[3[t.sup.3] - 5[t.sup.2] - 4]/[[t.sup.6] - 3[t.sup.4] + [t.sup.3] - 2[t.sup.2] + 2t - 1]

and computes its zeros and poles, with regard to their multiplicities and determines whether they lie in T, [T.sub.+], or [T.sub.-] (see Figure 3). The factorization index is computed as x = 2 - 4 = - 2. Since x [not equal to] 0 the output is

4 [member of] [sigma]([T.sub.{a,b}]) and 4 [member of] [sigma]([[??].sub.{a,b}]).

Example 3.2. Let us consider now the paired singular integral operators [T.sub.{a,b}] and [[??].sub.{a,b}] defined in (2.4) and (2.5) with rational scalar coefficients

a(t) = [t.sup.21] + 3it + 1 + i and b(t) = [t.sup.6] - i[t.sup.5] + 3[t.sup.2] + (1 - 3i)t.

We want to check if the complex number [lambda] = i belongs to the spectra of those operators.

The [ASpecPaired-Scalar] algorithm constructs the auxiliary function

r(t) = [[[t.sup.21] + 3it + 1]/[[t.sup.6] - i[t.sup.5] + 3[t.sup.2] + (1 - 3i)t - i]]

and computes its zeros and poles, with regard to their multiplicities and determines whether they lie in T, [T.sub.+], or [T.sub.-]. Since one of the poles of r lies in T (see Figure 4) the rational function is not factorable and the algorithm concludes that

i [member of] [sigma]([T.sub.{a,b}]) and i [member of] [sigma]([[??].sub.{a,b}]).

Example 3.3. Let us consider the paired singular integral operators [T.sub.{a,b}] and [[??].sub.{a,b}] defined in (2.4) and (2.5) with rational scalar coefficients

a(t) = [t.sup.9] + 5[t.sup.2] - 1 - i and b(t) = [t.sup.11] + [t.sup.5] + 2[t.sup.3] - (10 + i)[t.sup.2].

Let us check if the complex number [lambda] = 0 belongs to the spectra of these operators. The [ASpecPaired-Scalar] algorithm constructs the auxiliary function r(t) = [[[t.sup.9] + 5[t.sup.2] - 1 - i]/[[t.sup.11] + [t.sup.5] + 2[t.sup.3] - (10 + i)[t.sup.2]]]

and computes its zeros and poles, with regard to their multiplicities and determines whether they lie in T, [T.sub.+], or [T.sub.-] (see Figure 5). The factorization index is computed as [chi] = 2 - 2 = 0 and the algorithm concludes that

i [??] [sigma]([T.sub.{a,b}]) and 0 [??] [sigma]([[??].sub.{a,b}]).

4 Spectral algorithm: the rational matrix case

This section is dedicated to the formal description of the [ASpecPaired-Matrix] algorithm, that explores the spectra of paired singular integral operators, with rational matrix functions coefficients, defined on the unit circle.

4.1 [ARFact-Matrix] algorithm

The [ARFact-Matrix] source code, which can be found in [19], is used in the design of the spectral algorithm [ASpecPaired-Matrix] for paired singular integral operators [T.sub.{a,b}] and [[??].sub.{a,b}] defined in (2.4) and (2.5), with rational matrix functions coefficients, defined on the unit circle.

In [19] we present in detail the implementation of the complete and explicit rational matrix functions factorization algorithm [ARFact-Matrix]. This analytical algorithm, implemented using the computer algebra system Mathematica, computes explicit left and right factorizations of the form (2.6) for non-singular rational matrix function, defined on the unit circle.

Similar to the [ASpecPaired-Scalar] algorithm, the success of the [ARFact-Matrix] algorithm (and, consequently, of the [ASpecPaired-Matrix] algorithm) depends on the possibility of finding solutions of polynomial equations. However, due to the complexity of the matrix case, it is not as feasible as before to use the Root objects to obtain an explicit matrix function factorization when working with polynomials of a fifth and higher degree. In fact, one key step of this algorithm consists of finding the zeros of the determinant of the rational matrix function. This means that the size of the matrix function is also a limiting factor, even when its entries are rational functions with low degree polynomials.

We note that our description of the rational matrix function factorization algorithm (see Section 3 of [19]) is very different from other descriptions that can be found in the main literature, which are always presented from a more analytic perspective, rather than an algorithmic one. In particular, our description differs significantly from the description given in [43], which is the one that we followed more closely in our implementation. The main reason for these differences is that, on one hand, a computer, unlike most humans, has no problem performing iterated tasks like while-condition-not-satisfied-compute-function-number-i-and-replace-row-number-j-of-matrix-A. On the other hand, computers (and programmers) notably, still have a hard time dealing with instructions like given-function-f-that-satisfies-these-conditions-find-matrix-B-with-these-properties. As a consequence, in our implementation of the [ARFact-Matrix] algorithm we were compelled to make several significant changes, in order to render the algorithm more computer friendly, from a symbolic computation point of view.

We also note that, although the final factorization may have relatively simple entries, if we were to use the traditional pencil and paper tools the intermediate calculations would take typically many working hours for matrix sizes such as the ones used in examples of subsection 4.3 and higher, up to the point of infeasibility.

4.2 [ASpecPaired-Matrix] algorithm

In this subsection it is shown how the symbolic and numeric computation capabilities of the computer algebra system Mathematica were used to construct a spectral algorithm to explore the spectra of paired singular integral operators, with rational matrix functions coefficients, defined on the unit circle, of the form

[T.sub.{a,b}] = a[P.sub.+] + b[P.sub.-] and [[??].sub.{a,b}] = [P.sub.+]aI + [P.sub.-]bI,

defined in (2.4) and (2.5), respectively, with a,b [member of] [[R(T)].sub.n,n], n [??]2.

The [ASpecPaired-Matrix] algorithm checks if a complex number (chosen arbitrarily) belongs to the spectra of operators [T.sub.{a,b}] and [[??].sub.{a,b}]. As in the scalar case, the implementation of this spectral algorithm with Mathematica allows to obtain in a simpler and more immediate way the results of lengthy and complicated calculations. In the design of this spectral algorithm we used the factorization algorithm [ARFact-Matrix] (see [19]), that computes explicit factorizations for given factorable rational matrix function defined on the unit circle, as discussed in the previous subsection. For this reason, the success of the [ASpecPaired-Matrix] algorithm depends on the possibility of finding solutions of certain polynomial equations. In fact, one key step of this algorithm consists of finding the zeros of the determinant of a rational matrix function defined through the matrix functions a and b. This means that the size of the matrix is also a limiting factor, even when its entries are rational functions with polynomials with fourth or less degree.

The [ASpecPaired-Matrix] algorithm can be used to explore the spectra of a given paired singular integral operator of classes [T.sub.{a,b}] and [[??].sub.{a,b}] defined in (2.4) and (2.5). For each pair of input functions a, b [member of] [[R(T)].sub.n,n] and complex number [lambda], the [ASpecPaired-Matrix] algorithm gives the output

[[lambda] [member of] [sigma]([T.sub.{a,b}]) and [lambda] [member of] [sigma]([[??].sub.{a,b}])] or [[lambda] [??] [sigma]([T.sub.{a,b}]) and [lambda] [??] [sigma]([[??].sub.{a,b}])] or

[[lambda] [??] [sigma]([T.sub.{a,b}]) and [lambda] [member of] [sigma]([[??].sub.{a,b}])] or [[lambda] [member of] [sigma]([T.sub.{a,b}]) and [lambda] [??] [sigma]([[??].sub.{a,b}])].

Note that, due to the noncommutativity of matrix multiplication and the inclusion of [ARFact-Matrix] algorithm, the code of this spectral algorithm is much more elaborate than the code of [ASpecPaired-Scalar] algorithm.

In Figure 6 is shown the pseudo code of the [ASpecPaired-Matrix] algorithm.

To check whether a given complex number belongs to the spectrum of the paired singular integral operator the algorithm works as follows.

The spectral algorithm [ASpecPaired-Matrix] checks whether det(a - [lambda]e) [equivalent to] 0 or det(b-[lambda]e) [equivalent to] 0. If this happens then the output is that the complex number A belongs to the spectra of operators [T.sub.{a,b}] and [[??].sub.{a,b}]. Otherwise, if at least one pole of the entries of the matrix functions a or b lies in T, then the output is [[lambda] [member of] [sigma]([T.sub.{a,b}]) and [lambda] [member of] [sigma]([[??].sub.{a,b}])]. Else the algorithm constructs the auxiliary rational matrix function

r = [(a - [lambda]e).sup.-1](b - [lambda]e). (4.1)

If r is not a factorable matrix function (with respect to the unit circle), that is, if at least one zero of the determinant of r lies in T, then the output is [[lambda] [member of] [sigma]([T.sub.{a,b}]) and [lambda] [member of] [sigma]([[??].sub.{a,b}])]. If the rational function (4.1) admits a factorization of the form (2.6), then the auxiliary rational matrix function

[??] = (b - [lambda]e)[(a - [lambda]e).sup.-1] (4.2)

is constructed and the left partial indices [[chi].sub.i] and [[??].sub.i] of r and [??], respectively, are computed using the [ARFact-Matrix] algorithm.

With the knowledge of [[chi].sub.i] and [[??].sub.i], using Theorem 2.15, the spectral algorithm can infere if the complex number [lambda] is in the spectra of operators [T.sub.{a,b}] and [[??].sub.{a,b}]. The algorithm concludes that [lambda] [??] [sigma]([T.sub.{a,b}]) if [[chi].sub.i] = 0, [[for all].sub.i] = [bar.1,n]. Else, the conclusion is that [lambda] [member of] [sigma]([T.sub.{a,b}]). The algorithm also concludes that [lambda] [??] [sigma]([[??].sub.{a,b}]) if [[??].sub.i] = 0, [[for all].sub.i] = [bar.1,n]. Else, the conclusion is that [lambda] [member of] [sigma]([[??].sub.{a,b}]).

As in the scalar case, the analysis of the code reveals that a key step in this algorithm is the computation of the zeros and poles (with regard to their multiplicities), of the rational function (4.1) and whether they lie in T, [T.sub.+], or [T.sub.-].

4.3 [ASpecPaired-Matrix] examples

In this subsection we present nontrivial examples computed with the [ASpecPaired-Matrix] algorithm. The factors [r.sub.[+ or -]] and [[??].sub.[+ or -]], computed with the [ARFact-Matrix] algorithm, are given in Appendix A. Note that, although these factors are not used directly by the [ASpecPaired-Matrix] algorithm to study the spectra of operators [T.sub.{a,b}] and [[??].sub.{a,b}] defined in (2.4) and (2.5), they are necessary for the computation of the left partial indices [[chi].sub.i] and [[??].sub.i].

Example 4.1. Let us consider the paired singular integral operators [T.sub.{a,b}] and [[??].sub.{a,b}] defined in (2.4) and (2.5) with rational matrix coefficients

[mathematical expression not reproducible] (4.3)

Since in this case ab [not equal to] ba, then the equality (2.13) is not necessarily satisfied.

Example 4.1.1 Let us check if the complex number [lambda] = 0 belongs to the spectra of these operators. Since det(a(t) - [lambda]e) [equivalent to] 0, the output is

[0 [member of] [sigma]([T.sub.{a,b}]) and 0 [member of] [sigma]([[??].sub.{a,b}])].

Example 4.1.2 Let us now check if the complex number [lambda] = -1 belongs to the spectra of the operators [T.sub.{a,b}] and [[??].sub.{a,b}], with rational coefficients given by (4.3). There are no poles in the entries of matrix functions a and b lying in T and therefore, the [ASpecPaired-Matrix] algorithm constructs the auxiliary matrix function r(t) = [(a(t) + e).sup.-1](b(t) + e) = [??]

and computes the determinant of r, det(r) = -1. Since there are no zeros of det(r) that lie on T, the [ASpecPaired-Matrix] algorithm constructs

[??](t) = (b(t) + e)[(a(t) + e).sup.-1] = [??]

and then gives for the left partial indices of r and [??], [[chi].sub.1] = [[chi].sub.2] = 0 and [[??].sub.1] = 1 and [[??].sub.2] = -1, respectively. Thus, the algorithm concludes that

[-1 [??] [sigma]([T.sub.{a,b}]) and -1 [member of] [sigma]([[??].sub.{a,b}])].

Example 4.2. Let us consider the paired singular integral operators [T.sub.{a,b}] and [[??].sub.{a,b}] defined in (2.4) and (2.5) with rational matrix coefficients

[mathematical expression not reproducible].

In this case we also have ab [not equal to] ba and so, the equality (2.13) is not necessarily satisfied.

We want to check if the complex number [lambda] = -1 belongs to the spectra of the operators.

There are no poles in the entries of matrix functions a and b lying on T and the [ASpecPaired-Matrix] algorithm constructs the auxiliary matrix function

[mathematical expression not reproducible]

and computes the determinant of r, det(r) = -it(t + 3). Since there are no zeros of det(r) that lie on T, the [ASpecPaired-Matrix] algorithm constructs

[mathematical expression not reproducible]

and then gives for the left partial indices of r and [??], [[chi].sub.1] = 1, [[chi].sub.2] = [[chi].sub.3] = 0, [[??].sub.1] = 1, and [[??].sub.2] = [[??].sub.3] = 0, respectively. Thus, the algorithm concludes that

[-1 [member of] [sigma]([T.sub.{a,b}]) and -1 [member of] [sigma]([[??].sub.{a,b}])].

Example 4.3. Let us consider the paired singular integral operator [T.sub.{a,b}] defined in (2.4) with rational matrix coefficients

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

We want to check if the complex number [lambda] = 1 belongs to the spectra of this operator.

There are no poles in the entries of matrix functions a and b lying on T and the [ASpecPaired-Matrix] algorithm constructs the auxiliary matrix function

[mathematical expression not reproducible]

and computes the determinant of r, det(r) = [[i(1 + 2t)(1 + 3t)(2[t.sup.4] + 11[t.sup.2] + 5t + 10)]/[2[t.sup.2](4 + [t.sup.2])]]. Since there are no zeros of det(r) that lie on T, the [ASpecPaired-Matrix] algorithm computes the left partial indices of r, [[chi].sub.1] = [[chi].sub.2] = [[chi].sub.3] = [[chi].sub.4] = 0. Thus, the algorithm concludes that

1 [??] [sigma]([T.sub.{a,b}]).

Example 4.4. Let us consider the paired singular integral operator [T.sub.{a,b}] defined in (2.4), with rational matrix coefficients

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

We want to check now if the complex number [lambda] = i belongs to the spectra of this operator.

There are no poles of the entries of matrix functions a and b lying in T. Therefore, the [ASpecPaired-Matrix] algorithm constructs the auxiliary matrix function

[mathematical expression not reproducible]

and computes the determinant of r, det(r) = [[[t.sup.2] + (t+2i)t-1]/[[t.sup.2]]]. Since there are no zeros of det(r) that lie on T, the [ASpecPaired-Matrix] algorithm computes the left partial indices of r, [[chi].sub.1] = [[chi].sub.2] = [[chi].sub.3] = [[chi].sub.4] = [[chi].sub.5] = [[chi].sub.6] = 0, and [[chi].sub.7] = -1. Thus, the algorithm concludes that

i [member of] [sigma]([T.sub.{a,b}]).

5 Special classes of paired singular integral operators

This section is devoted to special classes of paired singular integral operators with essentially bounded coefficients, defined on the unit circle. The generalized factorization algorithm [AFact] (see [20]) for special classes of essentially bounded matrix functions, which can be used to explore the spectra of some particular classes of paired singular integral operators, with given rational matrix functions coefficients, is described in subsection 5.2. Some nontrivial examples are presented in subsection 5.3.

Let us now consider the special class of matrix functions (see, for instance, [15], [16], [17], [18], [20], [21], [22], [23], [38], [39], [41], and [42])

[mathematical expression not reproducible] (5.1)

where [phi] is an essentially bounded matrix function on the unit circle, that is, [phi] [member of] [[L.sub.[infinity]][(T)].sub.n,n], [phi]* is the Hermitian adjoint of [phi] and [gamma] [member of] C.

Although we have theoretical results for [phi] [member of] [[L.sub.[infinity]][(T)].sub.n,n], we can always assume, without loss of generality, that [phi] [member of] [[L+.sub.[infinity]][(T)].sub.n,n] (see, for instance, [17], [20], [21], and [22]).

5.1 On the spectra of special classes of paired singular integral operators

Based directly on the ideas and concepts that can be found in [20] we formulate new results that relate the spectra of operators [T.sub.{a,b}] and [[??].sub.{a,b}] defined in (2.4) and (2.5) with the spectra of a special class of self-adjoint singular integral operators. Due to their simplicity, all proofs of the stated theorems are omitted in this section.

Theorem 5.1. Let us consider [lambda] [member of] C a fixed arbitrary number such that the matrix function [(a-[lambda]e).sup.-1] (b-[lambda]e) belongs to class (5.1), for a given matrix function [phi] [member of] [[L.sub.[infinity]][(T)].sub.[n/2],[n/2]], and [gamma] [member of] C.

[lambda] [member of] [sigma]([T.sub.{a,b}]) [??] - [gamma] [member of] [sigma] ([P.sub.+][phi][P.sub.-][phi]*[P.sub.+]) (5.2)

Based on Remark 2.4 of [20] the following result can be formulated.

Theorem 5.2. Let us consider [gamma] [member of] C a fixed arbitrary number. If [(a-[lambda]e).sup.-1] (b-[lambda]e) is a matrix function of class (5.1), for a matrix function [gamma] [member of] [[L.sup.-.sub.[infinity]][(T)].sub.[n/2],[n/2]] such that

[[phi].sub.-]([infinity]) = 0 and [gamma] [member of] C \ {0}, then [lambda] [??] [sigma]([T.sub.{a,b}]).

Using Corollary 2.2 of [20], we get the following result.

Theorem 5.3. Let [gamma] [member of] C be a fixed arbitrary number. Moreover, let [(a-[lambda]e).sup.-1] (b-[lambda]e) be a matrix function of class (5.1), for a matrix function [phi] [member of] [[L+.sub.[infinity]][(T)].sub.[n/2],[n/2]] and

[gamma] [member of] C. If [gamma] [member of] C \ [R.sup.-.sub.0], then [gamma] [??] [sigma] ([T.sub.{a,b}]).

Remark 5.4. Similar results to the Theorems 5.1, 5.2, and 5.3 can be obtained for the case when the transpose of the matrix function [(b-[lambda]e).sup.-1] (a-[lambda]e) belongs to class (5.1).

Theorem 5.5. Consider [gamma] [member of] C a fixed arbitrary number such that the matrix function (b-[lambda]e)[(a-[lambda]e).sup.-1] belongs to class (5.1), for a matrix function [phi] [member of] [[L+.sub.[infinity]][(T)].sub.[n/2],[n/2]] and [gamma] [member of] C.

[lambda] [member of] [sigma]([[??].sub.{a,b}]) [??] - [gamma] [member of] [sigma] ([P.sub.+][phi][P.sub.-][phi]*[P.sub.+]) (5.3)

Theorem 5.6. Let [lambda] [member of] C be a fixed arbitrary number. If (b-[lambda]e)[(a-[lambda]e).sup.-1] is a matrix function of class (5.1), for a matrix function [phi] [member of] [[L.sup.-.sub.[infinity]][(T)].sub.[n/2],[n/2]] such that

[[phi].sub.-]([infinity]) = 0 and [gamma] [member of] C \ {0}, then [lambda] [member of] [sigma] ([[??].sub.{a,b}]).

Theorem 5.7. Let [lambda] [member of] C be a fixed arbitrary number. Moreover, let (b-[lambda]e)[(a-[lambda]e).sup.-1] be a matrix function of class (5.1), for a matrix function [phi] [member of] [[L.sub.[infinity]][(T)].sub.[n/2],[n/2]] and [gamma] [member of] C. If [gamma] [member of] C \ [R.sup.-.sub.0], then [lambda] [??] [sigma]([[??].sub.{a,b}]).

Remark 5.8. Results similar to Theorems 5.5, 5.6, and 5.7 can be obtained for the case when the transpose of the matrix function (a-[lambda]e)[(b-[lambda]e).sup.-1] belongs to class (5.1).

5.2 [AFact] algorithm

In general, it is possible to show that the study of the factorability of essentially bounded Hermitian second-order matrix functions with negative determinant can be reduced to the study of the factorability of matrix functions of the form (5.1) (see, for instance, [15] and [42]). In addition, a canonical generalized factorization (see, for instance, [18]) of matrix functions of the class (5.1) has applications in several scientific research areas (see, for instance, [1], [18], [21], [25], [38], and [41]). In [15], [17], [19], and [20] we have stated necessary and sufficient conditions for the existence of a canonical generalized factorization [A.sub.[gamma]]([phi]) = [A.sup.+.sub.[gamma]][A.sup.-.sub.[gamma]] and gave explicit formulas for the factors [A.sup.+.sub.[gamma]] and [A.sup.-.sub.[gamma]].

Let [H.sub.r,[theta]] (T) denote the set of all bounded and analytic functions in [T.sub.+] that can be represented as the product of a rational outer function r and an inner function [theta]

(i.e., [theta] is a bounded analytic function on the interior of the unit circle such that its modulus is equal to one a.e. on T). For the case when [phi] [member of] [H.sub.r,[theta]] (T), we designed the generalized factorization algorithm [AFact] (see [15] and [20]) that computes a left generalized factorization of factorable essentially bounded matrix functions of class (5.1), for any general inner function [theta].

In particular, the [AFact] algorithm allows us to check if a matrix function of class (5.1) admits, or not, a left generalized factorization of the form (2.6). Moreover, if [A.sub.[gamma]]([phi]) is factorable, the algorithm allows to determine whether the generalized factorization is canonical or non-canonical, and it gives an explicit left generalized factorization of the matrix function. As a consequence, based on Theorem 5.1, Theorem 5.5, and Theorem 2.1 of [20], the [AFact] algorithm allows to check if a complex number [lambda] (chosen arbitrarily) belongs to the spectra of the singular integral operators [T.sub.{a,b}] and [[??].sub.{a,b}] defined in (2.4) or (2.5), for the case when the matrix functions [(a-[lambda]e).sup.-1] (b-[lambda]e) or (b-[lambda]e)[(a-[lambda]e).sup.-1] belong to class (5.1), respectively. Note that the computations of the [AFact] algorithm do not depend on the degree of the polynomials that may eventually be part of inner function [theta]. Therefore, for some subclasses of operators [T.sub.{a,b}] and [[??].sub.{a,b}], whose spectra cannot be studied with the [ASpecPaired-Matrix] algorithm due to the many zeros and poles present in the entries of the corresponding matrix coefficients, it may still be possible to use [AFact] to perform this analysis.

5.3 Essentially bounded examples

In this subsection we present nontrivial examples related with the class of essentially bounded matrix functions of the form (5.1). We consider the paired singular integral operators [T.sub.{a,b}] and [[??].sub.{a,b}] defined in (2.4) and (2.5) with essentially bounded matrix coefficients

[mathematical expression not reproducible],

where (9) [phi] [member of] [L.sub.[infinity]] (T) and [gamma] [member of] C \ {0}.

Once more, in this general case, we have ab [not equal to] ba and therefore, the equality (2.13) is not necessarily satisfied.

Example 5.9. Let us check if the complex number [lambda] = i belongs to the spectra of these operators.

Since det(a(t) - [lambda]e) [equivalent to] 0, the output is

i [member of] [sigma]([T.sub.{a,b}]) and i [member of] [sigma]([[??].sub.{a,b}]).

Example 5.10. Let us now check if the complex number [lambda]=i-1 belongs to the spectra of operator [T.sub.{a,b}] defined in (2.4).

Based on Theorem 2.15 we define the matrix function

[mathematical expression not reproducible]

In this case, r(t) is a matrix function of class (5.1), and therefore Theorems 5.1, 5.2, and 5.3 can be used to check if A belongs to the spectrum of operator [T.sub.{a,b}].

Example 5.10.1 Let [phi] be an arbitrary essentially bounded function such that

[P.sub.-][phi]=[phi]. By Theorem 5.2, we can conclude that

i-1[??][T.sub.{a,b}]

Example 5.10.2 Let [phi] be an arbitrary essentially bounded function and [gamma] [member of] \ [R.sup.-.sub.0]. By Theorem 5.3, we can conclude that

i-1[??][sigma]([T.sub.{a,b}]).

Example 5.10.3 Let [phi] be an arbitrary essentially bounded function and [gamma] [member of] [R.sup.-] . In this case, the [AFact] algorithm can be used to check if [lambda] belongs to the spectrum of operator[T.sub.{a,b}] defined in (2.4).

Let us consider[phi](t)=[[theta](l)]/[t-2], where [theta] is an arbitrary inner function.

(i) Let [theta] be a differentiable inner function in a neighborhood of t = 1, defined on T, and [gamma] = -1. The factorization algorithm [AFact] finds out that the matrix function r(t) admits a canonical generalized factorization in [L.sub.2](T). Therefore, we can conclude, by Theorem 5.1, that

i - 1 [??] [sigma] ([T.sub.{a,b}]).

(ii) Let [theta] be a differentiable inner function in a neighborhood of t = -1, defined on T, such that satisfies the condition (10) 0'(--1) = 0 and [gamma] = -1/9. The [AFact] algorithm determines that the matrix function r(t) admits a non-canonical generalized factorization in [L.sub.2](T). Thus, we can conclude by Theorem 5.1 that

i-1 [member of] [sigma] ([T.sub.{a,b}]).

Example 5.11. Let us now check if the complex number [lambda] = i-1 belongs to the spectra of operator [[??].sub.{a,b}] defined in (2.5).

Using Theorem 2.15, we define the matrix function

[mathematical expression not reproducible]

Since it is possible to rewrite [??](t) as

[mathematical expression not reproducible]

the study of the factorability of this matrix function can be reduced to the study of the factorability of

[mathematical expression not reproducible]

which is a matrix function of class (5.1). From here, Theorem 5.7 can be used to conclude that

i-1 [??] [sigma] [[??].sub.{a,b}]

for all [phi] [member of] [L.sub.[infinity]](T) and [gamma] [member of] C \ {0}.

6 Final Remarks

The design of our analytical algorithms is focused on the possibility of implementing on a computer all, or a significant part, of the extensive symbolic and numeric calculations present in the algorithms. The methods developed so far rely on innovative techniques of Operator Theory and have a great potential to be extended to ever more involved and general problems. Also, by implementing these methods on a computer, new and powerful tools are created which allow to explore that same potential, making the results of lengthy and elaborate calculations available in a simple way to researchers of different areas.

* We hope that the work presented in our papers may give a significant contribution to the further development of the role of symbolic computation in the study of all, or at least some, of the problems in Operator Theory.

* We are also considering the design and implementation of other factorization, spectral and kernel algorithms. In particular, we hope to publish in the near future some results concerning algorithms that allow to explore the spectra and compute the kernels of singular integral operators related with Hankel and commutator operators.

* We note that the majority of the concepts and results established for the unit circle within Operator Theory can be generalized for the real line. It is our opinion that the design and implementation of analytical algorithms that work with singular integral operators defined on the real line can constitute a very interesting new line of research.

* We hope that our work within the Operator Theory, and with Mathematica, points the path to the future design and implementation of several other analytical algorithms, with numerous applications in many areas of research and technology.

* We also hope that, going forward, these analytical methods, and their implementation using a computer algebra system with large symbolic and numeric computation capabilities, may contribute to the development of the numerical approach in Operator Theory.

Acknowledgements

The authors wish to thank Professor Saitoh for the insightful discussions, that took place during their visits to Aveiro in recent years, on the use of the symbolic computation capabilities of Mathematica applied to several topics of Operator Theory.

Appendix A

Example 4.1.2

Left canonical factorization of r

[mathematical expression not reproducible]

Left non-canonical factorization of r

[mathematical expression not reproducible]

Example 4.2

Left non-canonical factorization of r

[mathematical expression not reproducible]

Left non-canonical factorization of [??]

[mathematical expression not reproducible]

Example 4.3

Left canonical factorization of r

[mathematical expression not reproducible]

Example 4.4

Left non-canonical factorization of r

[mathematical expression not reproducible]

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Ana C. Conceicao

Center for Functional Analysis, Linear Structures and Applications (CEAFEL)

Departamento de Matematica

Universidade do Algarve, Campus de Gambelas 8005-139, Faro, Portugal

E-mail: aicdoisg@gmail.com

Jose C. Pereira

Center for Environmental and Sustainability Research (CENSE)

Center for Functional Analysis, Linear Structures and Applications (CEAFEL)

Departamento de Engenharia Electronica e Informatica

Universidade do Algarve, Campus de Gambelas 8005-139, Faro, Portugal

(1) This research was partially supported by Centro de Analise Funcional e Aplicacoes (CEAF), Institute Superior Tecnico (Portugal), under FCT project PEst-OE/MAT/UI4032/2014

(2) All the research presented in this paper was done with Mathematica 9. At present time, we are using Mathematica 10, with no backward compatibility issues to report. For further information on the computer algebra system Mathematica visit the Wolfram's website at www.wolfram.com.

(3) A preliminary version of this algorithm was presented in the 1st International Conference on Algebraic and Symbolic Computation [45].

(4) The meeting (see Figure 1) devoted to the 70th birthday of Professor Saburou Saitoh included several presentations on topics related to the study of special classes of singular integral operators.

(5) This kind of algorithm is very important to the design of new spectral algorithms.

(6) An analogous relation can also be stated for the case when [a.sup.-1] [member of] [[[L.sub.[infinity]](T)].sub.n,n] changing, in (2.8), [P.sub.+]a[b.sup.-1] [P.sub.-] for [P.sub.-]b[a.sup.-1][P.sub.+] and [P.sub.-]a[b.sup.-1][P.sub.+] for [P.sub.+]b[a.sup.-1][P.sub.-], respectively.

(7) The corresponding eigenspaces of this spectrum are the kernel and the image of P.

(8) The pretty-print functionality allows to write on the computer screen scientific formulas in the traditional format, as if one was using pencil and paper.

(9) The overline in [bar.[phi]] denotes the complex conjugate of [phi], defined over the unit circle.

(10) This condition is provided explicitly in the output of the [AFact] algorithm. It arises from the construction of a homogeneous linear system which we know to be uniquely solvable when -[gamma] [member of] [sigma] ([P.sub.+][phi][P.sub.-][bar.[phi]][P.sub.+]).