Ijeart01012

International Journal of Engineering and Advanced Research Technology (IJEART) Volume-1, Issue-1, July 2015

On Mathematical Operator Systems and applications to information technology N. B. Okelo 

have

Abstract— The study of tensor products, operator systems and spectral theory of operators form a very important focal point in functional analysis. In this paper, we give results on properties of tensor products in Hilbert spaces of operator systems and subsystems.

Bs,i x  x1  ...  xi 1  Bs ,i xi  xi 1  ...  xn

and on all

the other elements of H  H  ...  H the operator  is Bs ,i 1 n defined on linearity and continuity.

Definition 3 ([5], [6]).

Let x  x  x  ...  x be an eigenvector of the 0,...,0

1

2

n

Index Terms—Resultant, Operator, Multiparameter System, Eigenvalue, Eigenvectors, Tensor products.

system (1), corresponding to its eigenvalue   ( ,  ,...,  ) ; 1 2 n then is m , m ,..., m - th associated vector (see[4]) to xm1 ...., mn 1 2 n

I. INTRODUCTION

an eigenvector x of the system (1) if there is a set of 0,0,...,0

. The method of separation of variables in many cases turns out to be the only acceptable, since it reduces finding a solution to a complex equation with many variables to find a solution to a system of ordinary differential equations, which are much easier to study. In this work we consider operator systems and their applications to ICT II. PRELIMINARIES We give some definitions and concepts from the theory of multiparameter operator systems necessary for understanding of the further considerations.

Let the linear multiparameter system be in the form:

Bk ( ) xk  ( B0, k   i Bi , k ) xk  0,

(1)

i 1

k  1, 2,..., n k ,i

act in the Hilbert space H

i

Definition 1. [1,2,11]   ( ,  ,...,  )  C n is an eigenvalue 1 2 n of the system (1) if there are non-zero elements x  H , i  1, 2,..., n such that (1) is satisfied, and i

i

decomposable tensor x  x  x  ...  x is called the 1 2 n eigenvector corresponding to eigenvalue   ( ,  ,...,  )  C n .

1

2

n

Definition 2. The operator

Bs,i

is induced by an

operator B , acting in the space H , into the tensor i s ,i space H  H  ...  H , if on each 1 n tensor x  x  ...  x of tensor product 1

space

x

i1 ,i2 ,...,iin

decomposable

n

H  H1  ...  H n

we

N. B. Okelo, School of Mathematics and Actuarial Science, Jaramogi Oginga Odinga University of Science and Technology, P. O. Box 210-40601, Bondo-Kenya

13

 H

1

  H n

, satisfying to conditions

B0, i ( ) xis , s2 ,,,, sn  B1,i xs1 1, s2 ,..., sn  ...  Bn,i xs1 ,..., sn1 , sn 1  0

xis1 , s2 ,..., sn  0 , when si  0

.

(2)

0  sr  mr , r  1, 2,..., n, i  1,..., n For element

the

indices

in s1 , s2 ,..., sn there are various

( xi1 ,i2 ,...,in )  H1   H n

arrangements

n

where operators B

vectors

from

set

of

, integers

on

n

with

0  sr  mr , r  1, 2,..., n, .

Definition 4. In [1,3,11] for the system (1) is an analogue of the Cramer’s determinants, when the number of equations is equal to the number of variables, and is defined as follows: On decomposable tensor x  x  ...  x operators  are 1

n

i

defined with help the matrices

 0   B0,1 x1 n  B0,2 x2  i  i x     i 0  B0,3 x3  ...  B x  0, n n

1

2

B1,1 x1 B1,2 x2 B1,3 x3 ...

B1, n xn

B2,1 x1 B2,2 x2 B2,3 x3 ...

B2, n

....  n   ... Bn,1 x1  ... Bn,2 x2   ... Bn,3 x3  ... ...   ... Bn, n 

(3)

where  ,  ,...,  are arbitrary complex numbers, under the 0 1 n expansion of the determinant means its formal expansion, when the element x  x  x  ...  x is the tensor products 1 2 n of elements x , x ,..., x If   1,  0, i  k , ,then right 1

2

n

k

i

side of (10) equal to  x , where x  x  x  ...  x On all 1 2 n k the other elements of the space H operators  are defined by i

linearity and continuity. E (s  1, 2,..., n) is the identity s

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On Mathematical Operator Systems and applications to information technology operator of the space H .Suppose that for all x  0 , i , and all B are selfadjoint (0 x, x)   ( x, x),   0 i,k operators in the space H . Inner product [.,.] is defined as i follows; if x  x  x  ...  x and y  y  y  ...  y are 1 2 n 1 2 n decomposable tensors, then [ x, y]  ( x, y) where ( x , y ) is 0

i

i

the inner product in the space. H .On all the other elements i of the space H the inner product is defined on linearity and continuity. In space H with such a metric all operators   1 are selfadjoin i

0

i

Definition5.( [7],[8]) Let two operator pencils depending on the same parameter and acting in, generally speaking, in various Hilbert spaces be as follows

B ( )  B i

1

A1  E2 ... . ...

0 ... A0  E2 E1  B1 ... . ... . ...E1  B0

An  E2 .

A1  E2 E1  Bm .

E1  B1

H1 , H 2 ,..., H n

)

H  H1  ...  H n are introduced

of the

  B0,1   0     0    B0,i   0     0 

 B1,1  0,1

B

  1,1

B 





Bk1 ,1  k1 1.1

B



 0,1

 B1,1

0

 B

B1,i



Bki ,i

B0, i

B1,i 



 0

 B0, i

 B1,i

0   B  0       Bk1 ,1  , 0.  0   Bki ,i  0      Bki ,i  

 k1 ,1

i  2,3,..., n

which acts in the ( H  H )n  m - direct sum of n  m copies of 1 2 the space H  H In a matrix (4), the number of rows with 1

product

i

Ri 1

...

0   ... .  ... An  E2   ..

0   ... .  .... E1  Bm 

n

tensor

a finite dimensional Hilbert space H , correspondingly. i Without loss ofcopies with

B( )  B0   B1   B2  ...   Bm

 A0  E2  .   0   E1  B0  .  . 

i  1, 2,..., n

operators R (i  1,..., n  1) with the help of operational i matrices (3.12) Let B ( ) be the operational bundles acting in

m

Operator Re s( A( ), B( )) is presented by the matrix

2

k1  k2 spaces

n

2

  B1,i  ...   ki Bki ,i ,

Bi ( ) - operator bundles acting in a finite dimensional Hilbert space correspondingly.

Suppose Hi that k  k  ...  k . In the space H k1  k2 (the direct sum of

A( )  A0   A1   A2  ...   An , 2

0,i

2

operators A is equal to leading degree of the parameter  in i pencils B( ) and the number of rows with B is equal to the i

leading degree of parameter  in A( ). The notion of abstract analog of resultant of two operator pencils is considered in the[7] for the case of the same leading degree of the parameter in both pencils and in the [2] for, generally speaking, different degree of the parameters in the operator pencils.

The number of rows with operators B , s  0,1,..., k in the s ,1 1 matrix R is equal to k and the number of rows with i 1 2 operators B , s  0,1,..., k is equal to k . We designate 1 s ,i i

 p  Bi ( )  the set of eigenvalues of an operator Bi ( ) .From [5] we have the result: Theorem 2.[9] n i 1

 p ( Bi ( ))   

if and only if

n 1 i 1

KerRi    , ( KerBk1  { }).

III. MAIN RESULTS

Consider the system Theorem1 [7,8].

Let for all operators bounded in corresponding Hilbert spaces, one of operators A or B has bounded inverse. n

m

Then operator pencils A( ) and B( ) have a common point of spectra if and only if

Ker Re s( A( ), B( ))   Remark 1. If the Hilbert spaces H and H are the finite 1 2 dimensional spaces then a common points of spectra of operator pencils A( ) and B( ) are their common eigenvalues. (see [6], [7].)

14

k1,s

kn , s

r 1

r 1

Ai , j , s ( ) xs  ( A0, s   1r A1, r , s  ...   nr An , r , s      ... Ai1 ,...,in i1 1

i2 2

in n

(4)

s  1, 2,..., n The parameters  ,  ,...,  enter the system nonlinearly, 1 2 n and the system (4) contains also the products of these parameters. Divide the system of equations (4) into groups of n in each group. If some equations remains outside, these equations we add by others operators from the system (4).

Each group contains n operators and will be considered separately.

In (4) the coefficients of the parameter

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International Journal of Engineering and Advanced Research Technology (IJEART) Volume-1, Issue-1, July 2015

mr , r  km , m  1, 2,..., n are the operators Ai , m, j , which act in the space H , index i indicate on the parameter  , index k i j - on the degree of the parameter  . i

We introduce the notations:

mr  k  k 1

2 ... km1  r

, r  km , m  1, 2,..., n

(5)

Further , we numerate the different products of variables 1 , 2 ,..., n in the system (4) on increasing of the

1  0 0  T1, s1 ,r   0 .  0 0 

0 0 ... 0 0 0  0   1 0 ... 0 0 0  0 0 0 1 ... 0 0 0    0 0 ... 0 0 0  ,… Tk 1, s ,r   0 . . . ... . . .    0 0 0 ... 0 0 0  0   0 0 ... 0 0 0 

T( s1 , s2 ,..., sn )r

0  1 0   0 .  0 0 

degrees of the parameter 1 . Let the numbers of term with the products of the parameters 1 , 2 ,..., n are equal to r Put further i i i 1i 2i ...ni  ( 1 2 ...n )t  k  k 1

1

n

2

n

2

1

2 ... kn  t

, tr,

where t  s is the number which correspond the multiplier at i1 i2 the ordering of multiplies of parameters in the 1 2 ...nin system (4). So in new notations to the product 1i1 2i2 ...nin correspond the parameter k1  k2 ... kn t , t  r (

1i i2 ...in  k  k ... k t , 1

n

2

1

2

n

tr

), aсcоrdingly, operators

1

1

rows of the matrix

T1, s1 , r ;

diagonal elements of the rows

2

also

to

1

and

so

T1, s1 , r ,..., Tki 1, si1 , r ,..., T( s1 , s2 ,..., sn )r

on. Besides, all have the order

(T2, n 1  1T0, n 1  2T1, n 1 ) xn 1  0

when s is the number of different products of parameters, entering the system(4).

In new notations the system (4) in the tensor product of spaces H  H  ...  H contains k  k  ...  k  s n

(8)

s1  s2  ...  sn .

Adding the system (7) by the following equations

(6)

Ak1 , k2 ..., kn ;i  Dk1  k2 ... km t ,i , t  1, 2,..., s;; i  1, 2,..., n

2

0 0 ... 0 0 1   0 0 ... 0 0 0  1 0 ... 0 0 0   0 1 ... 0 0 0  . . ... . . .   0 0 ... 1 0 0  0 0 ... 0 1 0 

The number 1 stands on the diagonal elements of the first s

is equal matrices

i  1, 2,..., n

1

n

s1  s2  ...  si 1  1,..., s1  s2  ...  si of the matrix Tk 1, s , r i i 1

Ar , s ,i  Dk1  k2 ... ks1  s ,i , r  1, 2,..., n; s  1, 2,..., kr ;

kr  max kr ,i , i  1, 2,..., k ,

n

0 0 ... 0 0 0   0 0 ... 0 0 0  0 0 ... 0 0 0   0 0 ... 0 0 0  . . ... . . .   0 0 ... 0 1 0  0 0 ... 0 0 1 

n

.......................................................... (k1  k2  2T2, n  k1  k2  2  k1  k2 1T0, n  k1  k2  2  k1  k2 T1, n  k1  k2  2 ) xn  k1  k2  2  0 ........................................................

parameters and n equations. Let k  k  ...  k  k Then 1 2 n n

kr

[ r  0 k 1

k1  k2 ... kr 1  k

(k1 ... kn1  2T

r

Dk1  k2 ... kr1  k ,i ]xi  [ k t Dk t ,i ]xi  0  0 k 1

2,1

n1

 ki

 k1 ... kn1 1,0T

0,1

i 1

n1

 ki

) xn  k1 ... kn1  2  0

i 1

(7)

k0  0; ki  0; i  1,2,..., n

(k1 ... kn  2T2  k1 ... kn 1T0  k T1 ) xk  0

Adding the system (7) with help of new equations so manner that the connections between the parameters, following from the equations of the system (4), satisfy.

Introduce the operators acting in the finite T0 , T1 , T2 , T0 , T 0 dimensional space R 2 and defining with help of the matrices

0 1 1 0  0 0 0 1 T0    , T1    , T2    , T0   , 1 0  0 0 0 1 0 0

(T0,t  1Ti1 ,t  k1 1Ti2 ,t  ...  k1 ... kn1 1Tin ,t  k  (i1 ,i2 ,...,in )t T(i1 ,i2 ,...,in ) ) xt  0 t  1, 2,..., r

Denote

k  (i

11 , i2 ,..., in )r

parameters, coefficient

15

(9)

xs  R 2 , s  n

entering

A(i11 ,i2 ,...,in )r

the multiplier

1i 2i ...ni of 1

2

n

the

the system (4) having the . System ((4),(9)) form the linear

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On Mathematical Operator Systems and applications to information technology multiparameter

system,

containing

k1  k2  ...  kn  r equations and k  k  ...  k  s parameters. To this system 1 2 n we may apply all results, given in the beginning of this paper.

T1, s1 , r Theorem 3. [4]. Let the following conditions: а) operators in the space H are bounded at Ak ,t , Ak1 , k2 ,...,, kn ;t i the all meanings i and k . b) operator  1 exists and bounded satisfy:

1  0 0   0 .  0 0 

0 0 ... 0 0 0   1 0 ... 0 0 0  0 1 ... 0 0 0   0 0 ... 0 0 0  ,…, . . ... . . .   0 0 ... 0 0 0  0 0 ... 0 0 0 

0

Then the system of eigen and associated vectors of (4) coincides with the system of eigen and associated vectors of each operators  (i  1, 2,..., n) i

Tk1 ... kn1 1, sn , r

Given two equations from (9) . Let the equations be:

(T2  1T0  2T1 ) xn 1  0

(10)

(1T2  2T0  3T1 ) xn  2  0

0  0 0   0 .  0 0 

Let 1  0 и xn 1  (1 , 1 )  0 is the component of the eigenvector of the system ((4),(9)). We have T( s1 ,..., sn )t , r

0 0 0 1 1 0    1    2    1 , 1   0 , 1 0 0 0 0 1

1 1 21  0 , 1  11  0 , 2  0; 2  12 .

Further

from

the

condition it

1  0, 2  0, xn  2  ( 2 , 2 )  0 follows       0,       0 and consequently, 2 2 3 2 1 2 2 1 , Consequently, 2 . Earlier we proved that 13  2 2  12 . 3  13 On analogy for other parameters of ((4),(9)): if is the eigenvalue of the system -((4), (1 , 2 ,..., k1  k2 ... kn  s )

For matrices

( 

0 ... 0

0

0

0 ... 0

0

0 0 .

0 0

0 0 .

0 0

0 0 .

0 0

... ... ... ... ...

0 0 .

0 0

0

...

0

0

0 1 0 .

0 0

0 0 1 .

0 0

... ... ... . ... ...

0 0 0 .

1 0

0 0 0 .

0 1

operators,

1  0 0  0 .  0 0  0  0 0  0 .  0 0 

defining

T1, s1 , k t , T2, s2 , k t ,..., Tn, sn , k t , T0.k t

with help the act in space R s1 ... sn

(1 ,...,  s1  s2 ... sn )  R s1 ... sn 1

n 1

k

.

T1 k1 ... kn1 , sn , r ) 

i 1

 k  t T( s1 ...snt r ) Consequently,

each rj2 j2

0

(T0, r  1T1, s1 , r  ...  

r  1, 2,..., n  1; s  1, 2,..., kn .

rj1 j1

0 0 ... 0 0 0   0 0 ... 0 0 0  0 0 ... 0 0 0   0 0 ... 0 0 0  . . ... . . .   0 0 ... 0 1 0  0 0 ... 0 0 1 

0

On eigenvector

(9)) , then    4 , …, , … , , k1  1k1 k1  k2 ... kr  s  rs1 4 1

To parameters

0  1 0   0 .  0 0 

T0, r

0  0 0   0 .  0 0 

multiplier of it is corresponded   )t  k t ; t  r rjk jk

11  k  t s ... s 1

n

...............................

1 s   s 1

the equation

1

1

k 1 s 1   s 1

1

1

(T0,t  k  1T1,i1 , k t  k1 1T2,i2 , k t  ...  k1  k2 ... kn1 1Tn,in , k t 

.......................

k  (i1 ,i2 ,...,in )t T(i1 ,...,in )t ,t  k ) xk  t  0

k 1 s  s   s  s 1

1

2

1

2 1

...........................



Consider the last equation, in which

1

n 1

 ki i 1

s s 1

2  ...  s1 n 1 1 ki

 i 1

  s1  s2 ... s 1

n 1

 ri

1

i 1

....................................................

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International Journal of Engineering and Advanced Research Technology (IJEART) Volume-1, Issue-1, July 2015 Hence,

1s 2s  ns  k  s ; s  r.

1

2

n

For the obtained linear multiparameter system we construct operator  on rule (3).

0 The condition means that operators Ker 01  

N. B. Okelo is amathematician and researcher who won the young scientist national award by AU-TWAS 2013. His area of interest is pure mathematics and application to engineering and computing.

i  01i are pair commute [2]. So operators  i act in finite dimensional space H and operators have not the k1  k2 ... kr 1 1 zero eigenvalues then for the eigenvalue of the system((4),(9)) (1 , 2 ,..., k1  k2 ... kn )

any

0 (2.47),(2.48)) from [7] it follows there is such eigen element z that the equalities,  z   z , i  1, 2,..., k  k  ...  k satisfy. For analogy i,s

1

i,s

2

n

conditions we obtain the analogy results for all groups. We have the several systems of operator polynomials in one parameter. We apply the results of [9] The system has the form

i , z  i , s o,i z …………………..

k1  k2 ... ki1 1,i zi  i , s o,i zi Theorem 4. Let the conditions of the theorem1 is fulfilled.

All operators  have inverse. Then the system(4) has the o ,i

common eigenvalue if and only if

Ker

(k1  k2 ... ki1 1,i  i , s o,i )  0 .

IV. APPLICATIONS TO IT

Tensor product is a very important technique used in solving problems of norms in Hilbert spaces. Norms are very important properties of operators and interesting studies have been directed on them with applications to information technology particularly in cyber security.

REFERENCES [1] [2] [3] [4] [5]

[6] [7] [8] [9]

Atkinson F.V. Multiparameter spectral theory.

Bull.Amer.Math.Soc.1968, 74, 1-27.

Browne P.J. Multiparameter spectral theory. Indiana Univ. Math. J, 24, 3, 1974.

Sleeman B. D. Multiparameter spectral theory in Hilbert space.

Pitnam Press, London, 1978, p.118.

Dzhabarzadeh R.M. Spectral theory of multiparameter system of operators in Hilbert space, Transactions of NAS of Azerbaijan, 1-2, 1999, 33-40.

Dzhabarzadeh R. M, Salmanova G. H. Multtiparameter system of operators, not linearly depending on parameters. American Journal of Mathematics and Mathematical Sciences. 2012, vol.1, No.2.p.39-45.

Dzhabarzadeh R.M. Spectral theory of two parameter s system in finite-dimensional space. Transactions of NAS of Azerbaijan, v. 3-4 1998, p.12-18.

Balinskii A.I. Generation of notions of Bezutiant and Resultant DAN of Ukr. SSR, ser.ph.-math and tech. of sciences, 1980,2. (in Russian).

Khayniq X. Abstract analog of Resultant for two polynomial bundles Functional analyses and its applications, 1977, 2, p.94-95.

Dzhabarzadeh R.M. On existence of common eigen value of some operator-bundles, that depends polynomial on parameter. Baku.

International Topology conference, 3-9 oct., 1987, Tez. 2, Baku, 1987, p.93.

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