On some application of algebraic quasinuclei to the determinant theory

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ON SOME APPLICATION OF ALGEBRAIC QUASINUCLEI TO THE DETERMINANT THEORY

Grażyna Ciecierska

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

grac@matman.uwm.edu.pl

Abstract. In the paper we apply the modified powers of algebraic quasinuclei to construc- tion of determinant systems for quasinuclear perturbations of Fredholm operators. Given two pairs (Ξ,Χ), (Ω ,Υ) of conjugate linear spaces, an algebraic quasinucleus

(Ω→Ξ Χ →Υ)

∈an ,

F and a determinant system for the Fredholm operator

(Ω→Ξ Χ →Υ)

∈op ,

S , we obtain algebraic formulas for terms of a determinant system for S +T_F.

Keywords: determinant system, Fredholm operator, quasinucleus, quasinuclear operator

Introduction

Determinant systems for operators acting in infinite dimensional Banach spaces provide important tools for solving linear equations. The determinant system for linear operator A gives full information on solving the equation Ax =y₀, where y0 belongs to the range of A. The Sikorski’s and Buraczewski’s formulas [1-3]

for the solution are generalizations of the famous Cramer’s rule for solving finite systems of linear equations.

The first theory of determinants in arbitrary Banach spaces was developed by A.F. Ruston [4] and A. Grothendieck [5] and another one by T. Leżański [6], R. Sikorski [1] and A. Buraczewski [2]. A general approach to the theory of determinants was proposed by A. Pietsch [7], I. Gohberg, S. Goldberg and N. Krupnik [8].

The study of determinant systems leads to the study of concrete classes of Fred- holm operators. In this approach we consider the class of quasinuclear perturbations of Fredholm operators. Algebraic quasinuclei play an important role in the theory of determinant systems; if ( )D is a determinant system for a Fredholm _n operator S and T is the quasinuclear operator determined by an algebraic _F quasinucleus F , then we can obtain effective formulas for a determinant system for the operator S +T_F in Banach spaces. The purpose of this paper is to give purely algebraic formulas for terms of the mentioned determinant system. The for-

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mulas were first given by Plemelj [9] for endomorphisms of the form I + , where T T is an integral endomorphism, in the space ^C[^a^,^b]. These formulas were obtained on the basis of the Fredholm theory of integral equations. They were modified by Smithies [10], also in the case of endomorphisms I + , where T is integral. T R. Sikorski [1] generalized the formulas over the endomorphisms I + , where T T is quasinuclear. A. Buraczewski [2] made further generalization of these formulas in the case of operators of the form S + , where S is a fixed Fredholm operator of T order zero and T is quasinuclear. Later contribution was made by D.H.U. Marchetti [11], who presented an alternative to Plemelj-Smithies formulas in the case of endomorphisms I + , where T belongs to the trace class of endomorphisms in T a separable Hilbert space. In this paper we generalize Plemelj-Smithies formulas over the operators of the form S + , where S is an arbitrary Fredholm operator T and T is quasinuclear. The result is formulated by means of the modified powers of quasinuclei.

1. Terminology and notation

We begin with a brief review on the terminology used in the determinant theory.

We follow the notation of [1-4].

Let (_Ξ^,_Χ), (_{Ω ,}_Υ ), (_Λ^,^Z) denote pairs of conjugate linear spaces over K

(K=Ror K=C). A bilinear functional A:Ω×Χ →K, whose value at a point

(ω,x)∈Ω×X is denoted by Axω , satisfying the condition ^ωÂx=^ω( ) (Âx = ^ωÂ)^x, where Ax ∈ and Y ωA∈Ξ, is called (_{Ξ ,}_Υ)-operator on Ω ×Χ; the space of all

(_{Ξ ,}_Υ)-operators on Ω ×Χ is denoted by op(Ω →Ξ,X →Y). For fixed non-zero elements x ∈₀ X, ω ∈₀ Ω, x₀⋅ω₀ denotes the bilinear functional on

Y

Ξ× , defined by ξ(x₀⋅ω₀)y=ξx₀⋅ω₀y for (ξ,y)∈ Ξ ×Y. An operator

( ,Y X)

op

B∈ Ξ →Ω → such that ABA =A, BAB =B is said to be a generalized inverse of an operator A∈op(Ω →Ξ,X →Y). The value of a (µ+^m)-linear functional D:Ξ^µ×Υ ^m→K, µ,m^∈N^∪{ }⁰ , at a point

(^ξ ^,K^, ^ξµ^,^y ^,K^, ^y_m)∈^Ξ^µ×^Υ ^m

1

1 is denoted by 









ym

, , y

, D ,

K K

1

1 ξµ

ξ . A (µ+^m)-

-linear functional D on Ξ^µ×Υ^m is said to be bi-skew symmetric if it is skew symmetric in variables from both Ξ , and Y. A (µ+^m)-linear functional

K :

D Ξ^µ×Υ ^m→ is said to be(Ω ,Χ)-functional on Ξ^µ×Υ^m, if for any fixed elements ξ ,K,ξ_i₋ ,ξ_i₊ ,K,ξ_µ∈Ξ

1 1

1 (i=1,K,µ), y ,K,y_m∈Υ

1 there exists an ele-

ment x_i∈Χ such that 







=  ⁻ ⁺

m i

i

i y , , y

, , , , , D , x

K K K

1

1 1

1 ξ ξ ξ ξµ

ξ ξ for every ξ ∈Ξ and

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for any fixed elements ξ₁,K,ξ_µ∈Ξ , y₁,K,y_j₋₁,y_j₊₁,K,y_m∈Υ (j=1,K,m) there exists an element ω ∈j Ω such that 









= 

+

− j m

j

j y , , y ,y,y , , y

, D ,

y

K K

K

1 1 1

1 ξµ

ω ξ

for every y∈Υ .

A sequence ( )Dn n_∈N_∪_{{ }}₀ is called a determinant system for operator

( ^,^X ^Y)

op

A∈ Ω→Ξ → , if for n∈ N∪{ }⁰

D is bi-skew symmetric n (Ω,Χ)- -functional on Ξ^µⁿ ×Υ^mⁿ where n,m m n

n

n =µ₀+ = ₀+

µ , min µ( ₀,m₀)=⁰, there exists r∈ N∪{ }⁰ such that D_r ≠0 and the following identities hold:

∑( )

= − +

+ 









⋅ 

−

=











 ⁿ

n n

m

j j j m

n j j m

n y , , y ,y , , y

, D ,

y y , , y

, , , D A

0 0 1 1

1 0

1

1 1

K K

K µ ξ ξµ

ξ ω ξ

ω ,

∑( )

=

+

−

+ 









⋅ 

−

=











 ⁿ

n n

i m

i i n

i i m

n y , , y

, , , , D ,

y x , , y , Ax

, D ,

µ

µ ξ ξ ξ ξ

ξ ξ ξ

0 1

1 1 0

1 0

1 1

K K K

K K

, where

( µ ) Υ ( ) Χ ω Ω

Ξ

ξ_i∈ i=1,K, _n ,y_j∈ j=1,K,m_n , x∈ , ∈ . The least r∈ N∪{ }⁰ , such that D_r ≠0 is called the order of determinant system ( )Dn n_∈N_∪_{{ }}₀ . The integer

0

0 m

µ − is called the index of determinant system ( )Dn n_∈N_∪_{{ }}₀ . If A ∈op(Ω→Ξ,Χ →Υ ) is a Fredholm operator of order ^r=^min{ⁿ′^,^m′} and index d =n′−m′, B∈op(Ξ →Ω,Υ →Χ) is a generalized inverse of A^,

{z₁,K,z_n }

′ , {^ς1,K,^ς }

′

m are complete systems of solutions of the homogenous equations Ax=0 and ωA=0, respectively, then the sequence ( )Dn n_∈N_∪_{{ }}₀ defined by the formula

,

y y

z z

By By

z z

By By

y , , y

, D ,

r m n m m

r m n

n r n n r

n n r m n r n n r

n n

n r

m n

r m n

r n n n

0 0

1

1 1

1

1 1

K K

M M

K K

M M

K K

′−

′ +

′

−

′ +

− ′ + ′

′− +

′

−

′ +

−

′ +

′− + =









ς ς

ξ ξ

for ξ_i∈Ξ (i=1,K,n+n′−r),y_j∈Υ (j=1,K,n+m′−r), is a determinant system for the operator A^.

A linear functional F:op(Ξ →Ω,Υ →Χ)→K is said to be an algebraic quasinucleus on op(_Ξ ^→_Ω,_Υ ^→_Χ), if there exists T_F^∈op(_Ω^→_Ξ,_Χ ^→_Υ)

such that F(x⋅ω =) ωT_Fx for (^ω,x)∈^Ω×^Χ. T is called a quasinuclear operator _F

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determined by F. The space of all algebraic quasinuclei on op(Ξ →Ω,Υ →Χ) is denoted by ^an(_Ω^→_Ξ^,^X ^→^Y). If ∈Υ ξ ∈Ξ

0

0 ,

y are fixed, then algebraic quasinucleus ξ₀⊗y₀∈an(Ω →Ξ,Χ →Υ ) such that (ξ₀⊗y₀)( )B =ξ₀By₀ for

(Ξ →ΩΥ →Χ)

∈op ,

B is called one-dimensional. Every finite sum ∑

=

⊗

n

i

i y

1

ξ of one-dimensional quasinuclei is called finitely dimensional quasinucleus. By the trace of an algebraic quasinucleus ^F^∈^an(Ξ ^→Ξ^,Χ ^→^X) we understand the number TrF =F( )I . For F∈an(Ω→Ξ,Χ →Υ ) and C∈op(Λ→Ω,Υ →Ζ)

we define CF∈an(Λ→Ξ,Χ →Ζ):

(CF)( )A =F(AC) for A∈op(Ξ →Λ,Ζ →Χ). (1) Let D be a bi-skew symmetric (Ω,Χ)-functional on Ξ^µ×Υ^m, µ,m∈N, and

(Ω →Ξ Χ →Υ)

∈an ,

F . Fixing all the variables ξ₂,Kξ_µ∈Ξ , ∈Υ ym

, , y K

2

and interpreting D as the function of variables ξ₁,y₁ only, we define (µ+m−2)- -linear functional Fξ₁y₁D on Ξ^µ⁻¹×Υ^m⁻¹ by

( )

F( )A

y , , y

, D ,

F

m

y =







 K K

2 2 1 1

µ ξ

ξ

ξ ,

where







= 

ym

, , y , y

, , D ,

Ay

K K

2 1

2 1 1

1

ξµ

ξ

ξ ξ for ξ₁∈Ξ, y₁∈Υ .

We can iterate the procedure k times, k=min{µ,m}, provided D F F F

, D F F , D

F _y _y _y _y _y _y

k

k 2 2 1 1

1 1 2 2 1

1 ξ ξ ξ ξ ξ

ξ K K

are (_Ω^,_Χ)-functionals [12]. By a reasoning similar to that in [6], since D

F

F y y

k

k τ τ1 τ1

τ ξ

ξ′ ′ K ′ ′ does not depend on the choice of permutation τ of integers k

, ,K

1 , we denote by F F D

times k

43 42

1K

−

the common value of all F y F y D

k

k τ τ1 τ1

τ ξ

ξ′ ′ K ′ ′ . We also use the notation suitable for the formulation and the proof of the main theorem of the paper. A matrix

[ ]

m j ij i

a M

≤

= ≤

11 µ over the field K , is denoted by

(^M ^,K^,^Mµ)^T

1 , where ^M_i₌[^a_i₁^,^K^,^a_im],(i=1,K,µ), i.e. M is the i-th row of _i M .

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2. Algebraic formulas for terms of determinant systems for quasinuclear perturbations of Fredholm operators

We present the theorem, which gives a generalization of Plemelj-Smithies formulas for operators of the form S + , where S is Fredholm and T T is quasinuclear.

Theorem. Let S∈op(Ω→Ξ,X →Y) be a Fredholm operator of order

{ⁿ^,^m}

min

r= ′ ′ , index d =n′−m′≥0 and determinant system ( )Dn n_∈N_∪_{{ }}₀ . Sup- pose that U^∈op(_Ξ ^→_Ω,_Υ ^→_Χ) is a generalized inverse of S and {z ,K,z_n_'}

1 ,

{ς ^,K^,ς_m_'}

1 are complete systems of solutions of the homogenous equations Sx=0 and ωS =0, respectively. Then for any ^F^∈^an(_Ω^→_Ξ^,_Χ ^→_Υ ) which determines

(_Ω^→_Ξ _Χ ^→_Υ )

∈

=T op ,

T_F the following formulas hold:

1 2 1

1 2

1 1

1 1 0

1 0 0 1

0 0 0

σ σ σ

σ

σ σ

K K

M M M

M M

K K

43 42 1^K

−

=

−

+

k k k n

n n

times k

T T

k T

k T D F

F n k (2)

for n,k^∈N^∪{ }⁰ , where

( )

[

^UT ^UF

]

⁽^m ^, ^k⁾

Tr ^m

m= ⁻¹ =1K

σ (3)

n

n D

T =⁰ and T is the _n^m (₂ⁿ+ⁿ^'+^m^'−₂^r)-linear functional (4)

=



 





−

′ +

−

′ +

r m n

r n m n

n y , , y

, T ,

K K

1

1 ξ

ξ

( ) ( ) ( ) ( )

( ) ( ) [ ( )]⁽ ⁾ [ ( )]⁽ ⁾

,

z UT

T z

UT T y

TU y

TU

z UT

T z

UT T y

TU y

TU

z UT z

UT Uy

UT

z UT z

UT Uy

UT

n i ' m i

' m r m n i m i

m

n i i

r m n i i

n i r n n i

r n n r m n i r n n i

r n n

n i i

r m n i i

' m r ' n n '

m r ' n n '

m r ' n n

r ' n n r

' n n r

' n n

r ' n n r

' n n r

' n

∑ n

′

−

′ +

′

− ′ + ′

′

−

′ +

−

′ +

−

′ +

−

′ +

−

′ +

′

−

′ +

+

− + +

− +

+

− + +

− +

=

ς ς

ξ ξ

K K

M M

K K

M M

K K

1 1

where for s=1,Km', t=1,Kn'

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( )

[ ]⁽ ⁾

( )





=

= =

+

− +

−

+

− + +

− +

m , i

if z UT

T

i z if

UT T

s r ' n n t

i s

s r ' n n t

i

s nn' r s

s r ' n n

1K 0 0

ς 1

ς

and ∑ is extended over all finite sequences of non-negative integers

' m r ' n

in

,

i K ₊ ₋ ₊

1 , such that i i m

' m r ' n

n =

+ +K ₊ ₋ ₊

1 ; i.e.

=









−

′ +

−

′ +

r m n

r n m n

n y, , y

, T ,

K K

1

1 ξ

ξ

( ) ( ) [ ( )]⁽ ⁾ [ ( )]⁽ ⁾

∑

= +

+ + ′− ′

− ′ +′ +

+

+ ′

′− + +

′− +











= 

m i

i nm r n

i m

i i

r n n i

n r d n

m r n n r

n n r

n n

z ,

, z / y ,

, y

UT T , , UT

T / UT ,

, D UT

K

K K

1

1 1

1

1 ξ ς ς

ξ

where

( ) ( ) [ ( )]⁽ ⁾ [ ( )]⁽ ⁾ ₌













− ′ + ′

′

−

′

+ ⁺ ^′⁻ ⁺^′⁻⁺ ⁺^′⁻⁺ ^′

n r

m n

i m

i i

r n n i

n y , , y /z , , z

UT T , , UT

T / UT ,

, D UT

m r n n r

n n r

n n

K K

1 1

1

1 ¹ ξ ς ¹ ς

ξ

( ) ( )

, m i

i y if

, , y

UT ,

,

D UT _n _n _r

r m n

i r n n i

n

r n

n  + + =









= 

′− +

′−

+ ⁺^′⁻

K K

K

1 1

1 ¹ ξ

ξ

and if i i m

r n n <

+ +K ₊ _′₋

1 , then

( ) ( ) [ ( )]⁽ ⁾ [ ( )]⁽ ⁾ ^det(^D( )^, ^,^D( ) ) ^,

z ,

, z / y ,

, y

UT T , , UT T / UT ,

,

D UT ⁿ _nⁿ_n _r _m ^T

n r

m n

i m i

i r n n i n

m r n n r

n n r

n n

′ +

−

′ +

′

−

′ +

− ′

+′ =









 +′− +′−+ +′−+ ^′′

K K

K

K K

1 1

1

1 ¹ ξ ς ¹ ς

ξ

where

( )

[

( ) ( ) ( ) ( ) n

]

i j i

j r m n i j i

j n

j UT Uy, , UT Uy , UT z, , UT z

D =ξ ^j Kξ ^j ₊ _′₋ ξ ^j Kξ ^j _′

1

1 (j=1,K,n+n′−r)

( )











=

′

−

′ + +

−

′

+ K 12K3

n r m n j j

n j r n

n y , , y , , ,

D 0 0

1 ς

ς if i_n₊_n_′₋_r₊_j=0 (j=1,K,m′),

( ) [ ( ) ( ) ( ) ( ) n]

i j i

j r m n i j i

j n

j r n

n TU y , , TU y , TUT z , , TUT z

D₊_′₋₊ =ς ⁿ⁺ⁿ^′⁻^r⁺^j ₁Kς ⁿ⁺ⁿ^′⁻^r⁺^j ₊ _′₋ ς ⁿ⁺ⁿ^′⁻^r⁺^j⁻¹ ₁Kς ⁿ⁺ⁿ^′⁻^r⁺^j⁻¹ _′ if i_n₊_n_′₋_r₊_j =1,K,m(j=1,K,m′).

Proof. We shall use an induction argument on k to establish the formulas. If k = 0^{, then}D =ⁿ Tⁿ⁰^{for any}n∈ N∪{ }⁰ _{. Let}k be any fixed positive integer.

Assume that for any n∈ N∪{ }⁰ the formulas (2) hold. For fixed n