On some property of the modified power of an algebraic nucleus

ON SOME PROPERTY OF THE MODIFIED POWER OF AN ALGEBRAIC NUCLEUS

*UDĪ\QD&LHFLHUVND

University of Warmia and Mazury in Olsztyn, grac@matman.uwm.edu.pl

Abstract. Given two pairs



;^,&



^,



^ȍ,8



of conjugate linear spaces, we show that the modified power of an algebraic nucleus preserves



^ȍ,X



- weak continuity of multilinear functionals. An application of the result in the determinant theory is also considered.

Introduction

Algebraic nuclei play an essential role in the theory of determinant systems [1- -4]. They allow to construct determinant systems for nuclear perturbations of Fred- holm operators, i.e. if

 

Dn nN‰_{^ `}0 is a determinant system for Fredholm operator



ȍ ^,X Y



^,

op

A o; o then one can obtain effective formulae for determinant system for AT_F, where ^Fan



^{ȍo;,X oY}



^{. The terms Dn, n N‰}

^ `

^0,

are, in particular, bi-skew symmetric multilinear



^ȍ,X



- weakly continuous func- tionals. It is well known that functionals F^pkD_nk, kN, are bi-skew symmetric.

We shall prove that the modified power of an algebraic nucleus transforms



^ȍ,X



^-

weakly continuous functionals into



^ȍ,X



- weakly continuous functionals. There- fore, in view of the result, F^pkD_nk are also



^ȍ,X



- weakly continuous functionals.

1. Terminology and notation

Let



;^,&



^,



ȍ^,8



denote pairs of conjugate linear spaces over the same real or complex field K. A bilinear functional A:ȍu& oK, whose value at a point

 

Z,^x ȍu^X is denoted by ZAx, satisfying the condition Z^Ax Z

   

^Ax Z^Ax, where AxY and ZA;, is called



;^,8



- operator on ȍu&. Let



^{X Y}



op :o ,; o denotes the space of



;^,8



- operators on :u&. Each



^{X Y}



op

A ȍo;, o can simultaneously be interpreted as a linear operator 8

& o :

A and as a linear operator ^A:ȍ^o;. For fixed non-zero elements

0 X,

x  Z₀;, x0˜Z0 denotes the bilinear functional on ;uY, defined by



^x0 Z0



^y [^x0 Z0^y

[ ˜ ˜ for

 

[^{,y };^uY. A linear functional

 

^K

op

F: ; oȍ^,8 o& o is called an algebraic nucleus, if there exists



o; & o8



op ȍ ,

T_F such that ^F

 

^x˜Z Z^TF^x for

 

Z,x ȍu&^. The opera- tor T_F is called a nuclear operator determined by F. The space of all algebraic nuclei on op



; o ,ȍ8 o&



is denoted by an



ȍo;^,X oY



^. The value of a



P^m



-linear functional D:;^Pu8^moK, P^,m N‰

^ `

^0, at a point



^[₁,, ^[P,^y₁,, ^ym



^;Pu^8m is denoted by . , ,

, ,

1

1 ¸¸¹·

¨¨©§

ym

D y



 [_P

[ A



^Pm



^-

linear functional D on ;^Pu8 ^m is said to be bi-skew symmetric if it is skew sym- metric in variables from both ;, ^{and Y. A}



^Pm



-linear functional

K

: u ^mo

D ;^P 8 is said to be



^ȍ,&



- weakly continuous functional on

m, 8

;^Pu if for any fixed elements [₁,,[_i1,[_i1,,[P;



^{i 1,,P}



^,

8

m y , ,

y1 there exists an element xi& such that

¸¸¹·

¨¨©§ _{ }

m i

i

i D y y

x , ,

, , , , , ,

1

1 1 1





 [ [ [ [_P

[ [ ^{for every} [; and for any fixed elements

, ,

1, [ ;

[  P ^y₁,,^yj_1,^yj_1,,^ym8



^j 1,,^m



there exists an element ȍ

j

Z ^{such that}

¸¸¹·

¨¨©§



 j m

j

jy D y y y y y

, , , , , ,

, ,

1 1 1

1





 [_P

Z [ ^{for every} y8^.

For Fan



: o;,& o8



and a bi-skew symmetric



ȍ^,&



-weakly conti- nuous functional D on ;^Pu8^m, interpreted as a function of variables [₁, y₁ ^only, we define a



^{P m2}



-linear functional F _y D

1

[1 on ;^P1u8^m1 by the formula

 

^,_, ^,_,

 

1^, 2

2 1

1 F A

y D y

F

m

y ¸¸¹·

¨¨©§



 _P

[ [ [

where

¸¸¹·

¨¨©§

ym

y D y

y

A , , ,

, , ,

2 1

2 1 1

1

1 

 [_P [

[ [

for [₁;,y₁8.

If ^{k min}

^ `

P^{, m,} then assuming that F _y F _y F _yD

k k k

k 1 1 [ 2 2 [11

[_{   }  is



ȍ^,&



- weakly continuous functional and interpreting it as a function of variables [_k,y_k only, we define a



P^{m 2} ^k



-linear functional F _y F _y F _yD

k k k

k [ 1 1 [11

[ _{ }  on

k m

k 

 u8

;^{P by}

 

_,^, _,^,

 

^,

1 1 1

1 1

1 k

m k

k y y

y F A

y D y

F F

Fk k k k ¸¸¹·

¨¨©§







 

 _[  ^P

[

[ [ [

where

 

_¨¨©^§ _¸¸¹^·













m k

k k k y y

y k

k

k y , y , , y

, , D ,

F F

F y

A k k k k 

 

1 1 1

1 2 2 1 1

P [

[

[ [ [ [

[ ^for [k;^,yk8 ^.

Since for fixed ^Fan



^{ȍo;,& o8}



and every permutation W of integers ,

, ,

1 k ₁₁

1

1y y y

y F F F

F[Wk Wk [W W [k k [ [3], the common value of all

1 W1 W W

W [

[ y F y

F

k

k  is

denoted by

  

times k

F F



‡

‡ [2]. Moreover, F^‡k denotes the modified k-th power of a nucleus F, i.e. 

times k

k F F

F k



‡

! ‡ p 1

.

2. Main theorem

Given ^Fan



^{ȍo;,X oY}



^{, Bop}



^{; oȍ,YoX}



^{, let} T TF and

 

^{BT  m B}

 

^{TB m for m N‰}

^ `

⁰ . Obviously,

 

^{BT  m op}



^{; oȍ,YoX}



^{. If}

2 1

1m  m

m , m1,m2 N‰

^ `

⁰ , then

 

^{BT  m y Fcyc}



^[c

 

^{BT  m1 y˜[}

 

^{BT  m2 yc}



[ _[ ^for

 

^{[,y ;uY (1)}

 

^{BT mx Fcxc}



^[c

 

^{BT m1x˜[}

 

^{BT  m2 xc}



[ _[ ^for

 

^{[,x ; uX (2)}

 

^{TB my F cyc}



^Zc

 

^{BT  m1 y˜Z}

 

^{TB m2yc}



Z _Z ^for

 

Z,^y ȍu^{Y (3)}

Lemma. Let Fan



ȍo;^,X oY



^, Bop



; oȍ^,YoX



^,

^

^{n, m}

`

^,

min

r c c nc^,mcN‰

^ `

^0, z₁,,z_ncX, 9₁,,9mcȍ. If nN andp



p1^,^,pn_n_cr



^{, q}



^q₁,,^qn^mc^r



are permutations of the integers

r n n c , ,

1 and 1,,nmcr, respectively, then for every integer

^

^{n n r n m r}

`

kd  c  c

d min ,

0

¸¸¹·

¨¨©§

–

–

–

^{ c  mc }

i q i n

i

i p r

n

i

q p y

y F_{k k i}By _{i nri}z y _nri

F

1 1

1

1

1 [ [ [ 9

[  (4)

 

 

–   –  

–

^{k i i i c k nki i nmcrknk i nki}

i

l i n

k r n n

i

i k n

i

m

k BT y BT z TB y

c

1 1

1

W V

W

V [ 9

[

for every



[1,^,[n_n_cr,y1,^,yn_m_cr



;^nncruY^nmcr, where c_k ,K

^

^,

`

^,

minn n r k n m r k

n_k d  c   c 

 

1^, nk i i

m

 

1 ^,

nk k r n n i i

k ^{ c  }

 

li ⁿi ₁^mcrknk are finite sequences of non-negative integers, V



V^1,^,Vⁿⁿc^r^k



^,



IJ , IJ,n m r k



IJ 1  c_ are permutations of integers k1,,nncr and ,

, ,

1 n m r

k   c respectively.

Proof. Induction on k



^{k 0, ,min}

^

^{nncr,nmcr}

` 

^{. Let}



1^{, , ,} 1^{, ,}



^.

r m n r n n r m n r

n

n_{ c} y y _{ c} ; ^{ c} uY ^{ c} [

[ ^{ } If k = 0, then (4) holds for c₀ 1,

0 n r,

n  m_i 0



^{i 1, ,nr}



^, ki ⁰



^{i 1, ,nc}



^, li ⁰



^{i 1, ,mc}



^,

.

, q

p W

V Suppose that (4) holds for k



^0dkmin

^

^{nncr,nmcr}

` 

^{. Then}

¸¸¹·

¨¨©§

–

–

–

^{ c  c }





m

i q i n

i

i p r

n

i

q p y

y F_{k k i}By _{i nri}z y _nri

F

1 1

1

1 1 1

1 [ [ [ 9

[ 

 

 

   

_¸^¸

¹

·

¨¨

©

§

–

^{ c}

–

^{     c}

–

^{   }





k

k i n i k

i k i

n k

i i i k

k

n k r m n

i

l i n

k r n n

i

i k n

i

m k

y c BT y BT z TB y

F

1 1

1

1

1 V W V W

[ [ [ 9

In the case: 1

0 k

Vi ^, Wi₀ ^k1, 1di0dnk, we obtain

 

 

   

_¸^¸

¹

·

¨¨

©

§

–

^{ c}

–

^{     c}

–

^{   }





k

k i n i k

i k i

n k

i i i k

k

n k r m n

i

l i n

k r n n

i

i k n

i

m k

y c BT y BT z TB y

F

1 1

1

1

1 V W V W

[ [ [ 9

 

 

   

 

   

˜

– –

–



 c





 c



z













k

k i n i k

i k i

n

k

i i i i

k k

n k r m n

i

l i n

k r n n

i

i k

n

i i i

m k

m k

y k

y TB z

BT

y BT y

BT F

c

1 1

1 1 1

0 0

1 1

W V

W V

[

9 [

[ [

 

 

–   –  

–

^{ c    c  }

z

  

k

k i n i k

i k i

n k

i i i

n k r m n

i

l i n

k r n n

i

i k n

i i i

m

k BT y BT z TB y

c

1 1

1 1

0

W V

W

V [ 9

[

where ^{ck1 ckF}

>  

^{BT  mi0}

@

If 1,

1 k

Vi Wi₂ k^{1, 1}di1^,i2 dnk^, i₁zi₂, then according to (1)

 

 

   

¸¸¹

·

¨¨©

§

–

^{ c}

–

^{   }

–

^{c  }





k

k i n i k

i k i

n k

i i i k

k

n k r m n

i

l i n

k r n n

i

i k n

i

m k

y c BT y BT z TB y

F

1 1

1

1

1 V W V W

[ [ [ 9

 

 

 

 

  _–  

 

z



 ˜





k

i i i i

i i i k

k

n

i i i i

m k

m m

k y

kF BT y BT y BT y

c

2 1 2

2 1 1 1

1

, 1 1

1 W V V W

[ [ [ [

  –  

–

^{   c  }

c







k

k i n i k

i k i

n

n k r m n

i

l i n

k r n n

i

i

k z TB y

BT

1 1

W

V 9

[

 

 

   

^,

1 1

, 1 1

2 1

–

–

–

^{c   c }

z

 _{ }

k

k i n i k

i k i n k

i i i

n k r m n

i

l i n

k r n n

i

i k n

i i i i

m

k BT y BT z TB y

c [_{V W} [_V 9 _W

where 1 ₂

 

^{1 2 1} _i1^.

i i

i BT y

c

c_k _k[_V ^{m m } _W

If 1,

1 k

Vi _{n i2} k^1,

W k ¹di1dnk^, 1di₂dnmcrkn_k, then by (3)

 

 

   

_¸^¸

¹

·

¨¨

©

§

–

^{ c}

–

^{     c}

–

^{   }





k

k i n i k

i k i

n k

i i i k

k

n k r m n

i

l i n

k r n n

i

i k n

i

m k

y c BT y BT z TB y

F

1 1

1

1

1 V W V W

[ [ [ 9

 

 

 

  _–  

 

z



 ˜





k

i i i i

i i k

k

n

i i i

m k

l i m

k y

kF BT y TB y BT y

c

1 2

1 2 1 1

1

1 1

1 W V W

[ [ 9 [

  –  

–

^{ c  }

z



 c







k

k i n i k

i k i

n

n k r m n

i i i

l i n

k r n n

i

i

k z TB y

BT

2 1 1

W

V 9

[

  –  

 

–  

^{ c}

–

^{  }

 

z





 c



z









k

k i n i k

i k i n k

i i i i i i

n k r m n

i i i

l i n

k r n n

i

i k n

i i i

m l

m i

k TB y BT y BT z TB y

c

2 1

1 2 1 2

1 1

1 1

W V

W V

W [ [ 9

9 .

If 1,

1 

_i k

n_k

V Wi₂ k^{1, 1}di1dnncrknk^, 1di₂ dn_k, then by (2)

 

 

   

¸¸¹

·

¨¨©

§

–

^{ c}

–

^{   }

–

^{c  }





k

k i n i k

i k i

n k

i i i k

k

n k r m n

i

l i n

k r n n

i

i k n

i

m k

y c BT y BT z TB y

F

1 1

1

1

1 V W V W

[ [ [ 9

   

 

  _–  

 

z



 ˜





k

i i i i

i i k

k

n

i i i

m k

m i

k k

y

kF BT z BT y BT y

c

2 2

1 2 1 1

1

1 1

1 V V W

[ [ [ [

  –  

–

^{   c  }

c



z





k

k i n i k

i k i

n

n k r m n

i

l i n

k r n n

i i i

i

k z TB y

BT

1 1

1

W

V 9

[

 

 

–   –  

–

^{ c  }

z



 c



z

 _{ }

k

k i n i k

i k i

n k

i i i

n k r m n

i i i

l i n

k r n n

i

i k n

i i i

m

k BT y BT z TB y

c

2 1

1 1

1

1 [V W [V 9 W ,

where 2

 

^{1 2} ₁^.

1

1 c TB ^{i i} y _i

c_{k k}9_i ^{m l } _W

If 1,

1 

_i k

n_k

V _{n i2} k^1,

W k ¹di1dnncrknk^,

,

1di₂dnmcrkn_k then we obtain

 

 

   

_¸^¸

¹

·

¨¨

©

§

–

^{ c}

–

^{     c}

–

^{   }





k

k i n i k

i k i

n k

i i i k

k

n k r m n

i

l i n

k r n n

i

i k n

i

m k

y c BT y BT z TB y

F

1 1

1

1

1 V W V W

[ [ [ 9

   



^{ ˜ }

 _–  

 





k

i i i i

i k

k

n

i

m k

l i i k k y

kF BT z TB y BT y

c

1 1

1 1 2 ²

1 1

1 V W

[ [ 9 [

  –  

–

^{ c  }

z



 c



z





k

k i n i k

i k i

n

n k r m n

i i i

l i n

k r n n

i i i

i

k z TB y

BT

2 1

1 1

W

V 9

[