# Cauchy-Binet type formulas for Fredholm operators

### Grażyna Ciecierska

Journal of Applied Mathematics and Computational Mechanics |
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CAUCHY-BINET TYPE FORMULAS FOR FREDHOLM OPERATORS

Grażyna Ciecierska

Faculty of Mathematics and Computer
Science, University of Warmia and Mazury in Olsztyn

Olsztyn, Poland

grac@matman.uwm.edu.pl

Received: 19 September 2016; accepted: 4 June 2017

**Abstract.**
Suppose , are Fredholm
operators acting in linear
spaces. By referring to the correspondence between
Fredholm operators and their determinant systems, we derive the formulas for a
determinant system for *AB *which are expressed via determinant systems
for *A* and *B*. In our approach, applying results of the theory of
determinant systems plays the crucial role and yields Cauchy-Binet type
formulas. The formulas are utilized in many branches of applied science and
engineering.

*MSC 2010:** 47A53, 15A15*

*Keywords: **Fredholm operator,
determinant system, reflexive generalized inverse, Cauchy-
-Binet theorem*

1. Introduction

The purpose of this paper is to exhibit a method of construction of a determinant system for a product of arbitrary linear Fredholm operators acting between linear spaces. The method is based on tools of the determinant theory created by Leżański [1], developed and modified by Sikorski [2-4] and Buraczewski [5, 6].

We address the
problem of how to express a determinant system for product *AB *of
Fredholm operators and ,
*X*, *Y*, *Z* being linear spaces over the same field (real or
complex), in terms of determinant systems for *A *and *B*.
In the derivation of the main result we use some ideas presented in [7] for
Fredholm endomorphisms and extend them to Fredholm operators acting between
arbitrary linear spaces. Since the method proposed in the paper is purely
algebraic, we dispense with assumptions related to a topological structure of
linear spaces involved. The formulas, obtained as a direct and constructive
solution to the above mentioned problem, are generalizations of the classical
Cauchy-Binet formula
[8-10], which states that if *A *and *B* are two matrices over field *F
*of sizes and ,
respectively, with , then ,
where the sum is taken over all increasing sequences ,
with , and () is submatrix
of *A* (*B*) obtained by deleting all columns (rows)
except these with indices in *p*. When , the
formula becomes the well-known product formula for
determinants. The Cauchy-Binet formula plays an important role in studies of
determinants, permanents and other classes of matrix functions. An increasing
interest in its applications in many branches of applied science, such as
matrix analysis and engineering [11-13], is a motivation of the paper. It
is worth emphasizing that, so far, many considerable contributions to
generalizing the Cauchy-Binet theorem have been made [14-17]. In our approach,
the proposed generalization to Fredholm operators is based on the
correspondence between any Fredholm operator and its determinant system. We
also make use of analogues of the Laplace expansion formula that are available
for terms of determinant systems.

2. Preliminaries

In this section we recall the main notions and facts concerning the determinant systems theory and we fix the notation [3-6, 18-20].

Suppose , and are
pairs of conjugate linear spaces (over the real or complex field *F*) with
respect to scalar products *I *on , *J*
on and *K *on ,
respectively, satisfying the cancellation laws [6]. Elements and are
called *orthogonal *if ; moreover, and for
given subsets and .
Denote by the value of
a -linear functional at a point .

*D* is said to be *bi-skew symmetric *if it is skew symmetric both
in variables and ; stands for the set of all bi-skew
symmetric functionals on . We call *D* an
* - functional on * if for arbitrary fixed elements and there
exists an element such that for every and for arbitrary fixed elements , there
exists an element such that for every . is
identified with the set of all - functionals on . A bilinear * -
*functional *D *on is said to be an *operator
on * and stands
for its value at . We denote by the set of all * - *operators on . Each can
be simultaneously interpreted as a linear mapping and
as a linear mapping . Thus for
. The operator ,
, being
fixed non-zero elements, defined by for , is called *one-dimensional*.

For let
, , , . *A*
is said to be a *Fredholm operator on of order
and index , *if
, , and [5,
21]. An operator satisfying
identities , is called
a *reflexive generalized inverse of * [22].
A sequence is said to be a *determinant system
for A* if , with , , , and the generalized Laplace expansion
formulas hold

,

where , , . The least , such
that , and the difference are called the* order *and the*
index of *, respectively.

As well-known [3, 5], an operator has a determinant system if and only if *A *is Fredholm;
the orders (the indices) of *A *and are
the same. Moreover, if *A* is Fredholm, is its reflexive generalized inverse and ,
* *form complete systems of solutions of the homogenous equations and ,
respectively, then defined by the formula

(1) |

for , is a determinant system for . |

3. Main result

In this section we examine Fredholm operators acting from one linear space into another one. We provide a construction of a determinant system for a product of two fixed Fredholm operators. For the sake of completeness, we start by quoting some auxiliary results concerning reflexive generalized inverses of Fredholm operators, which are necessary for the proof of the main theorem of the paper.

In what follows, , denote Fredholm operators of orders , and
indices ,
, respectively. Let , , and be
bases of , , and , respectively. The following direct sum decompositions hold: , being
subspaces such as and .
Moreover, denoting , , we
also obtain , where , and
. Let ,* *be bases of subspaces, , respectively, and , , where , being
the Kronecker symbol. Furthermore,
, being
subspaces of dimensions , , respectively.

Under the above given assumptions we recall [20] the following two results.

**Lemma 3.1.** *If
**, ** are arbitrary
reflexive generalized inverses of Fredholm operators **,**, respectively,
then ** and ** are bases of ** and **, respectively.*

* *

**Lemma 3.2.** *We assume that:*

(*i*) , * are
fixed determinant systems for Fredholm opera**tors
, , respectively**;*

(*ii*) * ** are such elements that:*

*;*

(*iii*) , *are such elements
that:*

*;*

(*iv*) * is a
reflexive generalized inverse of ** **defined by the formula*

*for*
*;* (2)

(*v*) * is a
reflexive generalized inverse of ** **defined by the formula*

*for*
*.* (3)

*Then
the operator is a reflexive generalized
inverse of .*

The following lemma plays an essential role in the sequel. It describes the connection between two arbitrary reflexive generalized inverses of a fixed Fredholm operator.

**Lemma 3.3.** *If ** are
reflexive generalized inverses of a Fredholm operator and ,** are bases of , **, respectively, then there exist elements ** **, ** ** such
that*

. | (4) |

**Proof.** By the relationship between *A *and its reflexive generalized
inverse *B* [5], there exist elements , * *such that * *,* *and the following identities hold:

, . | (5) |

Similarly, in view of the relationship
between A and C, |

, , | (6) |

where , are
elements satisfying conditions:
* *,* *. By (5) and (6), bearing in mind
that , . Consequently, ,
which implies

. | (7) |

It follows from (7) that |

. | (8) |

Since , we transform the right-hand side of (8) into the form

(9) |

.

Furthermore, remembering that , we
express (9) by |

. | (10) |

Hence, the identities , , combined with (8) and (10), lead to

. | (11) |

Finally, by putting and in (11), we arrive at (4), which is the required result.

Having established Lemmas 3.1-3.3, we are now in a position to state and prove the main result of the paper.

**Theorem 3.4.** *Let *, *b**e
** fixed determinant systems for Fredholm
operators ,, respectively.
If **, ** **are arbitrary reflexive generalized
inverses of **,**, respectively, then** the sequence ** ** defined by the formula:*

, | (12) |

for ,
where, are
arbitrary permutations of integers and , respectively, such that |

* **, **,*

is a
determinant system for. |

**Proof. **Let be reflexive generalized inverses of ,, respectively, defined
by formulas (2), (3). According to Lemma 3.2, * *is a reflexive generalized inverse of operator . It follows from (1), in view of Lemma
3.1, that the sequence defined by

(13) |

for , is a determinant system for . The order and the index of are equal to and , respectively. Let denote the reflexive generalized inverse of expressed by |

, | (14) |

where . Since and , . Let be an arbitrary fixed reflexive generalized inverse of . Hence, by Lemma 3.3, |

, | (15) |

for some . Assume . The orthogonality of to all and the orthogonality of to all , combined with (14), (15), lead to where is a finitely dimensional operator of the form |

, |

for some . Moreover, by virtue of (15), for any |

. |

Similarly, by (14), for any |

. |

Replacing by and by in (13), we conclude that the right-hand side of (13) remains unchanged. Furthermore, applying the formula for the generalized expansion of a determinant, we transform the determinant in (13) into the sum |

(16) |

multiplied by , where is a permutation of integers fulfilling the condition . Moreover, denoting
by q any permutation of integers such that and making use of
well-known properties of classical determinants, in view of (16), the
right-hand side of (13) is equal, up to a sign, to the sum |

(17) |

Next, taking into account the identities , , and , the sum (17) can be expressed (up to a sign) by |

(18) |

It follows from (18), bearing in mind the definition of and relying on properties of partitioned matrices, that the value is equal, up to a factor of 1 or , to |

(19) |

By combining (18), (19) with the relationship between and its determinant system, we give rise to the identity |

(20) |

where or . In view of (14), Lemma 3.3 implies that

for some , is an arbitrary reflexive generalized inverse of . Bearing in mind the bi-skew symmetry of , we can substitute for in (20). Since a determinant system for the fixed Fredholm operator is determined up to a constant (non-zero) factor, the sequence defined by (12) is a determinant system for . This completes the proof.

As a direct consequence of Theorem 3.4, we obtain the following result.

**Corollary 3.5.** *Under the assumptions of Theorem 3.4, with *, the *formula *(12)* is of
Cauchy-Binet type.*

4. Conclusions

In the paper,
products of Fredholm operators acting between arbitrary linear spaces were
considered. By exploiting terms of determinant systems for operators *A *and
*B*, with *AB*
well-defined, we provided a
direct construction of a determinant system for *AB*. The obtained result
leads to a generalization of the Cauchy-Binet formula to Fredholm operators and
yields an important tool for solutions of problems in various branches of
applied science and engineering.

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