Cauchy-Binet type formulas for Fredholm operators

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@article{Ciecierska_2017,
   doi = {10.17512/jamcm.2017.2.04},
   url = {https://doi.org/10.17512/jamcm.2017.2.04},
   year = 2017,
   publisher = {The Publishing Office of Czestochowa University of Technology},
   volume = {16},
   number = {2},
   pages = {43--54},
   author = {Grażyna Ciecierska},
   title = {Cauchy-Binet type formulas for Fredholm operators},
   journal = {Journal of Applied Mathematics and Computational Mechanics}
}

TY  - JOUR
DO  - 10.17512/jamcm.2017.2.04
UR  - https://doi.org/10.17512/jamcm.2017.2.04
TI  - Cauchy-Binet type formulas for Fredholm operators
T2  - Journal of Applied Mathematics and Computational Mechanics
JA  - J Appl Math Comput Mech
AU  - Ciecierska, Grażyna
PY  - 2017
PB  - The Publishing Office of Czestochowa University of Technology
SP  - 43
EP  - 54
IS  - 2
VL  - 16
SN  - 2299-9965
SN  - 2353-0588
ER  - 

Ciecierska, G. (2017). Cauchy-Binet type formulas for Fredholm operators. Journal of Applied Mathematics and Computational Mechanics, 16(2), 43-54. doi:10.17512/jamcm.2017.2.04

Ciecierska, G., 2017. Cauchy-Binet type formulas for Fredholm operators. Journal of Applied Mathematics and Computational Mechanics, 16(2), pp.43-54. Available at: https://doi.org/10.17512/jamcm.2017.2.04

[1]G. Ciecierska, "Cauchy-Binet type formulas for Fredholm operators," Journal of Applied Mathematics and Computational Mechanics, vol. 16, no. 2, pp. 43-54, 2017.

Ciecierska, Grażyna. "Cauchy-Binet type formulas for Fredholm operators." Journal of Applied Mathematics and Computational Mechanics 16.2 (2017): 43-54. CrossRef. Web.

1. Ciecierska G. Cauchy-Binet type formulas for Fredholm operators. Journal of Applied Mathematics and Computational Mechanics. The Publishing Office of Czestochowa University of Technology; 2017;16(2):43-54. Available from: https://doi.org/10.17512/jamcm.2017.2.04

Ciecierska, Grażyna. "Cauchy-Binet type formulas for Fredholm operators." Journal of Applied Mathematics and Computational Mechanics 16, no. 2 (2017): 43-54. doi:10.17512/jamcm.2017.2.04

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 operators , , 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)

, ,

(6)

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

.

(7)

.

(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 , be 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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