Existence of Necessary Condition for Normal Solution Operator Equation

IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 2 Ver. V (Mar. - Apr. 2017), PP 40-46 www.iosrjournals.org

Existence of Necessary Condition for Normal Solution Operator Equation ¹Md Najmul Hoda*, ²Mohammad Abid Ansari 1

Research Scholar* Department of Mathematics, T.M. Bhagalpur University, Bhagalpur-812007, India 2 Associate Professor in Mathematics T.N.B. College, Bhagalpur, T.M. Bhagalpur University, Bhagalpur– 812007, India E-mail:najmul85hoda@gmail.com Corresponding author*

Abstract: An operator means a bounded linear operator on Hilbert span. This paper proves the assertion made in its title. Following theorem yields the famous result AB + BA* = I = A*B + BA (1) Where A and B are the bounded linear operator on a Hilbert span H. where B* is self adjoint satisfying the above equation. After modification of this equation some interesting results are obtained.

Theorem: If AB + BA* = I has solution B, then

0 

(Re B).

Proof. If



(A) denotes the approximate point spectrum of A then

conjugate. Let 0 such that Now

 or,

 as

0    (A) and 0    B  . Further B 1  2 A .

   A , then 0     A*  ,

   A*      A

and so there exist a sequence

and

bar is complex

 X n of unit vector in H

A* xn  0 as n   .

AB + BA* = I

1   xn , xn  

 AB  BA  x , x  *

1   ABxn , xn    BA xn , xn 

n

n

*

 Bx , A x    A x , Bx   0 *

n

n  .

*

n

n

n

0     A .

Similarly we can prove that 0    B . since B 0    B Next we suppose that x  H be arbitrary then 2 x   AB  BA*  x, x This is contradiction, hence



 

 ABx, x    BA* x, x 

is a self adjoint, we have

  B    B

Hence



 Bx, A x    A x, Bx  *

*

 Bx A* x  A* x Bx

 i.e.

2 Bx Ax 1 2 B A

DOI: 10.9790/5728-1302054046

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Existence Of Necessary Condition For Normal Solution Operator Eqution i.e.

B 1  2 A

0    (Re A), this means that 0    (Re A), hence there exists a sequence xn  of unit vectors in H such that Re  A xn  0 as n   Now we suppose that

we have



2I  2  xn , xn  

AB  BA*  A* B  BA A  A* B  B A  A* xn , xn











2  2  Re ABxn , xn    B Re Axn , xn   or, I   Bxn ,Re Axn    Re Axn , Bxn   0 n as This is contradiction, hence 0    (Re A). Since (Re A) is self adjoint, hence   (Re A)=  (Re A).  0    Re A or,

x  H / 0 ). Then there is an inner product on

Theorem : If (i) has a solution B > 0 (i.e. (Bx, x) > 0 for all

H, equivalent to the inner product (.,.), such that Re A > 0 with respect to it.

Proof. We define a new equivalent inner product on H by

x, y   x, By  ,for all x, y  H

Since

 or,

 B, A  A*   0 we have B  A  A*    A  A*  B

 B  A  A  x, x    A  A  Bx, x   A  A  x, Bx    Bx,  A  A x     A  A  x, Bx  *

*

*

*

*

or,

 A  A  x, x   A

or,

Ax, x  A* x, x  A* x, x  Ax, x

or,

Ax, x  Ax, x  A* x, x  A* x, x

or,

*

*

 A  x, x

Re Ax, x  Re A* x, x

(2)

Again from the equation (1) we have

 A  A  B  B  A  A  then for any x  H .

2  x, x     A  A  Bx, x    B  A  A  x, x  2 || x ||   Bx,  A  A x     A  A  x, Bx     A  A x, Bx     A  A  x, Bx    A  A  x, x   A  A  x , x 2I 

*

*

*

Or,

2

*

*

*

*

*

*

DOI: 10.9790/5728-1302054046

*

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Existence Of Necessary Condition For Normal Solution Operator Eqution



Ax  x  A* x, x  Ax, x  A* x, x



Ax, x  Ax, x  A* x, x  A* x  x

 2Re Ax, x  2Re A* x, x  By  2  

 2Re Ax, x  2Re Ax, x  4Re Ax, x 1 Re Ax, x  || x ||2  0 2 Re A  0

Or,

xH

for all

Hence Theorem : If there exists a solutions A to (1) such that the eigen vectors of A* span H, then

W  B   R / 0 .

Proof. Let

   P  A* 

(* the point spectrum of A*) and let

corresponding to  . So we have From (1) we have

0  0 

or , or , or , or , or , Or,

A* x   x

for all

x H, x0

xH .

 A  A  Bx, x    B  A  A x, x   Bx,  A  A x    A  A x, Bx  *

*

*

*

 Bx,  x  Ax     x  Ax, Bx  0   Bx,  x    Bx, Ax     x, Bx    Ax, Bx  0    Bx, x    Bx, Ax     Bx, x    Ax, Bx  0   Bx, Ax    Bx, Ax         Bx, x  Re  Bx, Ax    Re   Bx, x  0 

Again we have

2 || x ||2

be an eigen vector

     

(3)

 A  A  Bx, x    B  A  A x, x   Bx,  A  A x    A  A x, Bx  *

*

*

*

 Bx,  x  Ax     x  Ax, Bx   Bx,  x    Bx, Ax     x, Bx    Ax, Bx    Bx, x     Bx, x    Bx, Ax    Bx, Ax 

      Bx, x   2Re  Bx, Ax  

2  Re   Bx, x   Re   Bx, x 

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Existence Of Necessary Condition For Normal Solution Operator Eqution

or ,

2 || x ||2

 4  Re   Bx, x 

or ,

1

 2  Re   By, y 

So that

|| y ||  1. Clearly

where y 

Re   0

x || x ||

This implies that

 By, y 



1 for all y  H 2  Re  

W  B   R / 0.

Hence

Theorem: If there exists a solution A to (1) such that the eigen vectors of A span H, then

  P  A and let 0  x  H xH

Proof. Let

be an eigen vector corresponding to

W  B   R / 0.

 . Then we have Ax = x

for all From equation (1) we have

0      or , Or,

 A  A  Bx, x    B  A  A x, x   Bx,  A  A x    A  A x, Bx  *

*

*

*

 Bx, A x   x    A x   x, Bx   Bx, A x     Bx, x    A x, Bx     x, Bx   Bx, A x    Bx, A x     Bx, x     Bx, x        Bx, x    Bx, A x    Bx, A x  Re   Bx, x   Re  Bx, A x  *

*

*

*

*

*

*

*

*

Again we have

or , 2 || x ||2  or , 2 || x ||2     

(4)

 A  A  Bx, x    B  A  A x, x   Bx,  A  A x    A  A x, Bx  *

*

*

*

 Bx, A x   x    A x   x, Bx   Bx, A x     Bx, x    A x, Bx     x, Bx        Bx, x    Bx, A x    Bx, A x  2Re   Bx, x   2Re  Bx, A x  *

*

*

*

*

*

*

or , 2 || x ||2  2 Re   Bx, x   Re   Bx, x 

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Existence Of Necessary Condition For Normal Solution Operator Eqution

or , 2 || x ||2  4Re   Bx, x   2Re   By, y 

or , 1

where y 

x || x ||

1 ,  Re   0  2Re  This show that W  B   R / 0.

 By, y  



Theorem : If there exists a solution B to equation (1) such that the eigen vectors of B span H, then either

W  A  R / 0 orW  Re A  R / 0.

Proof. Let   P  B  and let 0  x  H be an eigen vector corresponding to  . Then Bx   x all x  H .

for

From equation is (1) we have

0   

 A  A  Bx, x    B  A  A x, x   Bx,  A  A x    A  A x, Bx    x,  A  A  x     A  A  x ,  x  *

*

*

*

*

*

   x, A* x    x, Ax      A* x, x    Ax, x      Ax, x   Ax, x     Ax, x   Ax, x           Ax, x   Ax, x      2I m    2I m  Ax, x  





 









 4  I m    I m  Ax, x    Either I m  Ax, x   0 In this case Or

 Ax, x    Ax, x 

(5)

Im   0In thiscase   

(6)

Again from (1) we have



 A  A  Bx, x    B  A  A  x, x   Bx,  A  A x    A  A x, Bx    x,  A  A  x     A  A  x ,  x 



  x, A* x    x, Ax      A* x, x    Ax, x  



  Ax, x   Ax  x     Ax, x   Ax  x 

2 || x ||2  

*

*

*

*

*

*



DOI: 10.9790/5728-1302054046











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Existence Of Necessary Condition For Normal Solution Operator Eqution

      Ax, x    Ax, x 



(7)

Now two cases arise Case I : When

 Ax, x    Ax, x  then by (7)

We have

2 || x ||2 

4  Re   Ax, x 



2  Re   Ay, y 



l

where y 

Re   0. This implies that 1 for all y  H .  Ay, y   2  Re  

x || x ||

Obviously by

Hence

W  A  R / 0.

Case II :

   ; i.e.  is purely real from equation (7) we have 2 || x ||2  2  Ax, x   Ax, x     2 2Re  Ax, x  When

 

|| x ||2



1







4 Re  Ax, x 

2 Re  Ax, x 

 2 Re  Ay, y  where y  

Re  Ay, y  



z  z i.e.

x || x ||

1 for all y  H 2

(8)

We have

2a i.e.

Re  z  



1 zz 2

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

a



1 zz 2



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Existence Of Necessary Condition For Normal Solution Operator Eqution

     













1  Ay, y   Ay, y 2 1  Ay, y   y, A* y 2 1  Ay, y    A* y, y  2 1 A  A*  y, y  2 1  *   A  A  y, y  2   Re  A y, y 

 





(9)

Hence from (8) and (9) we have

  Re A y, y  

W  Re A   R / 0



1 for all y  H 2

References [1].

[2].

[3].

[4].

[5].

[6].

[7].

[8].

[9].

[10].

[11].

[12].

[13].

[14].

[15].

[16].

[17].

[18].

[19].

[20].

[21].

[22].

[23].

Simmons, G.F.: Introduction to topology and modern analysis. McGraw-Hill, New York (1963).

Rudin, Walter: Functional analysis (1981).

Duggal B.P. and Khalagai J.M.: On operator equation AB + BA* = A*B + BA = I, Indian J. Pure Appli. Math., 13(H), 1376-1383 November 1983.

Duggal B.P. and Khalagai J.M.: On operator equation AB + B*A = A*B + BA = I, Math Japan, 26(1981) R. Nakamato: On the operator equation THT = K Math. Japan; 18 (1973), PP. 251-252.

G.K. Pederson, M. Takesaki: The operator equation THT = K Proc. Amer. Math. Soc., 36 (1972), PP. 311-312.

T. Kato: Perturbation theory for linear operator, 2nd Edition, Springer, 1980.

Taylor, A.E.: Introduction to functional analysis, Wiley, New York, 1958.

Md. N. Hoda and Mohammad A. Ansari: Some normal solution on operator equation, AIJRSTEM (USA); 16(1); Sep. – Nov. (2016); PP. 49-51.

R. Meise, D. Vogt: Introduction to functional analysis, Oxford G.T.M.2, Oxford University Press 1997.

M.H. Mortad: On some product of two unbounded self adjoint operator, Integral Equation Operator Theory 64 (2009), 399-408.

M.H. Mortad: On Normality of the solution two Normal operators, complex Analysis Oper. Theory, 61 (2012) 105-112, DOI: 10.1007/s11785-010-0072-7.

M.H. Mortad: Products of unbounded normal operator, (submitted). arXIV: 1202.6143V1.

C. Putram: Commutation properties of Hilbert span operators, Springer, 1967.

W. Rudin: Functional analysis McGraw-Hill, 1991 (2nd edition) Schwanberg, Allen: The operator equation AX – XB = C with normal A and B, Pasifie J. Math., Vol. 102:447-453 (1982).

Read, Michael and Sincon, Barry: Functional analysis Academic Press, New York (1972).

Roth, W.E.: The equation AX – YB = C and AX – XB = C in matrices, Proc. Amer. Math. Soc., 5:392-396 (1952).

Rosen-blum, M.: On operator equation BX – XA = C Duk Math. J., 25, 263-269 (1956).

Rosen-blum, M.: The operator equation BX – XA = Q with self adjoint. A and B. Proc. Amer. Math. Soc., 20:115-120 (1969).

Bellman, R.: Introduction to matrix analysis, McGraw-Hill, New York (1960).

Barberian, S.K.: Lectures in functional analysis and operator theory, Springer-Verlag, New York (1979).

Barberian, S.K.: The numerical range of normal operator, Duke Math. J., 31:479-483.

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