1


International Journal of Analysis and Applications  

ISSN 2291-8639 

Volume 4, Number 1 (2014), 11-20 

http://www.etamaths.com 

 

 

 

_________ 

2010 Mathematics Subject Classification. 49M15. 

Key words and phrases. hod, approximation, operators, convexity. 

 

© 2014 Authors retain the copyrights of their papers, and all  open Access 

 articles are distributed under the terms of the Creative Commons Attribution License. 

11 

APPLICATIONS 

 
OCTAV OLTEANU 

 
Abstract. This review  work presents the general statemen t of a  

its applications. We consider the estimation of the absolu te error 

too. On e makes the connec tion to the contracti on principle. One 

of the applications is approximating ,1,
/1

pA
p

 where A a  

positive s elfadjoint operator is acting on  a Hilbert space. One 

tone operators. For the local  

approach, we mention appropriate references.  

 

 

1. Introduction 

 

 ([1], [4]). 

The evaluat a version 

for convex monotone operators. This statement and its proof are 

important due to the generality of the obtained result. One applies this 

method to equations involving functions of matrices or of operators, as well 

to the successive 

approximation method from the contraction principle. In particular, one 



12 

OLTEANU 

approximates ARppA
p

,,1,
/1

  being a selfadjoint operator with the 

spectrum contained in the positive semi axis. This work is related to the 

papers [2], [5], [6]. For some other aspects on the same subject (involving 

analyticity), see [1]. The background is contained in [3] and [4]. The rest of the 

paper is organized as follows. Section 2 contains the detailed proof of the 

In Section 3, some direct applications to concrete scalar and operatorial 

equations are considered. Section 4 makes the connection to the contraction 

principle, by approximating ApA
p

,1,
/1

  being a positive selfadjoint 

operator acting on a Hilbert space. 

 

 

2. General statements 

 

Let X be a   complete vector lattice, endowed with a solid 

 yxyx   and o  continuous norm  .0 xxxx
norderinn

 

Let Y  be a normed vector space, endowed with an order relation defined by a 

closed convex cone. For ,,, baXba   we denote    .,, bxaXxba   Pet 

  .,,
1

YbaCP   In most of our applications, we have ,YX   where Y  is an 

order complete Banach lattice of selfadjoint operators, that is also a 

commutative algebra (see [3], p. 303-305). 

 

Theorem 2.1 Assume additionally that for each       XYLxPbax ,,,
1




  and 

that 

 
     .bPxPaPbxa   

If     ,0,0  bPaP  then there exists a unique solution 


x  of the equation   ,0xP  

where 

      .,,:,liminf:
1

10
NkxPxPxxbxxxx

kkkkkk







           (1) 

Moreover, we have 



13 

 

     .0,
1




kk
xPaPxxbxa                         (2) 

Proof. Using induction upon ,k  we prove that 

  .,,0
1

NkxxxP
kkk




                                     (3) 

The last relations (1) and the convexity of ,P  yield 

                0,0
110


 kkkkkkk

xPxPxxxPxPxPbPxP  

Hence   .0 NkxP
k

  This relations lead to 

      .0
1

1

1
NkxxxPxPxx

kkkkkk







 

We derive the following useful relations 

 

           

        .,

00,0

1

1

NkaxxaxaxPxP

xPaPxPxPaP

kkkkk

kkk









 

 

Using the hypothesis on the space ,X  there exists 


x  defined by the first 

relations (1), and from (3) we infer that the sequence  
kk

x  is decreasing. 

Passing through the limit in the recurrence relations (1) one obtains 

        .00
1







xPxPxP  

From the assumptions on the positivity of    ,, aPbP   and from the  

definition of ,


x  we infer that .bxa 


 In order to prove (2), one uses the 

convexity once more: 

 

           

            .,0
11

NkxPaPxxxxxPaP

xxaPxxxPxPxPxP

kkkk

kkkk









 

The uniqueness of the solution follows quite easily: 

              .00
21212

1

221221





 xxxxxPxPxxxPxPxP  

Similarly, we can write: ,0
12


 xx  hence .21


 xx                                                      

□ 

 

The corresponding statement for convex decreasing operators holds.  



14 

OLTEANU 

Theorem 2.2 Assume that for any  bax ,  there exists   
1

 xP  such that 

           .,
111 




 bPxPbxaXYxP  If     ,0,0  bPaP  

Then there exists an unique solution [,] bax 
  of the equation   ,0xP  


x  being 

given by: 

      .,,,limsup
1

10
NkxPxPxxaxxxx

kkkkkk






  

Moreover, the sequence  
kk

x  is increasing and the convergence rate is given by the 

inequalities 

     .,
1

NkxPbPxx
kk




 

 

 

3. Direct applications 

 

During this Section we mention some applications of the general theorems 

of Section 2. The difficulties consists only in technical details concerning 

verifying conditions from general theorems. That is why we do not prove all 

the statements. 

 

Theorem 3.1 (see Theorem 2.4 [6]). Let H  be a Hilbert space and X  the 

commutative algebra defined in [3], p. 303-305. Let 

  0,0,,...,1,0,
0


 nj

BBnjXB  be such that 






n

j

j
BB

1

0  

and 1
21

2



n

n
UnBUBB   is invertible for any  .,0 IU   Then there exists a 

unique ,0, IUU   such that 

,0
01

1

1





BUBUBUB

n

n

n

n
  

and this solution verifies in particular the relation 






n

j

j
BBUI

1

0
 



15 

 

Proof. The space X  is an order-complete Banach lattice and a commutative 

algebra of selfadjoint operators. Let 

  XXIP 


,0:  

be defined by 

  .
01

BUBUBUP
n

n
   

 

One can show that the operator  
n

n
UUP   is convex on 


X  (see [6] for 

details). Now it follows easily that P  is also convex on ,X  since all the 

coefficients 


 XB
k

 and all the operators in X  are permutable. On the other 

hand, we have: 

 

    

        .,0,
~

,
~

2
~

2

1

12

11

12

1

IUXVVBUBUnBVUP

VBUBUnBVUP

n

n

n

n













 

 

We also have 

 

    

   
       .,00

00,0

11

1

1

1

IUIPUPP

VBnBVBUnB

VBVPIUUP

n

n

n











  

 

Because of the hypothesis, one has 

 

    ,0,00
0

1

0
 



BBIPBP

n

k

k
 

 

so that all requirements of Theorem 2.1 are accomplished. It follows that there 

exists a unique solution [,0] IU   of the equation   ,0UP  that verifies 

relation (9) for :0k  

     .0
01

1

1

1
BBBBIPPUI

n



  



16 

OLTEANU 

Now the proof is complete.                                                                                              
□ 

 

Theorem 3.2 Let XH ,  be as in the preceding theorem, XB  ,1  such that the 

spectrum     [.,ln]  BS  There is a unique solution XIU  [,0]  of the equation 

  0exp  IBU   

and this solution verifies in particular the relation 

.exp
1

IBBUI 


 

 

The next result is an application of the scalar version of Theorem 2.2, 

when .RYX   

 

Proposition 3.1 Let 0,,   be such that 

  .1exp1    

Then there exists a unique solution [1,0]


x  of the equation 

  0exp   xx  

and we have 

 
.1,0

exp

1
0 












x  

 

Theorem 3.3 Let H  be a finite dimensional Hilbert space, and X  the space defined 

in [3], p. 303-305. Let XCBA ,,
~

 be such the spectrums of A
~

 and B  are contained 

in [.,0]   Assume also that 

  .
~

exp ICBA   

 

Then there exists a unique solution [,0] IU   of the equation 

  0
~

exp  CBUUA  

and the following estimation holds: 

   ICCIBAAU 


,0
~

exp
~~ 1

 

 



17 

 

Proposition 3.2 There is a unique solution [2,2/3]


x  of the equation 

,0142
23

 xx  

and this solution verifies 

.2
152

27
2 


x  

 

Theorem 3.4 Let A  be a selfadjoint operator acting on a Hilbert space, with the 

spectrum    .2,2/3AS  Let  AXYX   defined in [3], p. 303-305. Then there 

exists a unique operator XU   such that: 

042
23

 IUU  

 

and this operator verifies 

  [2,
152

27
2] US  

 

 

4. Approximating ;1,
/1

pA
p  connection to contraction principle 

 

Let H be a Hilbert space, A  a selfadjoint operator acting on ,H  with the 

spectrum 

  )([,,0] AXXAS   the associated commutative algebra according to [3], p. 

303-305. We denote 

 




hAhhAh
hAhA

,sup,,inf
11

  

 

Theorem 4.1 (see Theorem 2.1 [2]). Let A  be as above,   .,1, RppISpA   

There exists a unique operator [,]
/1/1

IIU
p

A

p

Ap
   such that 

,0 AU
p

p  

and this solution verifies the relations 

 



18 

OLTEANU 

 

 

.,
1 11

/1

/1

/1

/1 







 AI

p
IUAI

p

UI
A

pp

Ap

ApApp

A

p

p

A




 

 

Corollary 4.1 With the above notations and assumptions, we have 

 

.
1

lnln
11 

 AIAI
AAA

A

AA



  

 

Remark 4.1  

 

         xPxPxxxx
kk

1

1
:,




   

the mapping   is a contraction, the rate of co nvergence of the sequence  
kk

x  

is given by contraction principle. Next, we show that this is the case of the 

operator   ,AUUP
p
  which leads to the positive solution .

/1 p

p
AU   

 

Theorem 4.2 (see [2]). Let XAp ,,  be as above. Then the Newton recurrence for the 

equation  

  0 AUUP
p  

is 

  .,
11

,
1

1

/1

0
NkAU

p
U

p

p
UUIU

p

kkkk

p

A








  

The convergence rate for p
k

AU
/1

  is given by 

.,
1 1/1/1

NkAI
p

p
AU

A

p

A

k

p

k














 



 

Proof. P  is: 

 

   

  .,
1111

,

111

1
1

1

/1

0

NkUAU
p

U
p

p
AU

p
U

p
U

AUpUUUIbU

k

p

kk

p

k

pp

kk

p

k

p

kkk

p

A

















 



19 

 

Let 

 ,;: /1 pAUXUM   
 

where the root is obtained by the aid of functional calculus for .A  Clearly, M  

is closed in ,X  hence it is complete. Let XM :  be defined by 

 

  .,
11 1

MUAU
p

U
p

p
U

p






  

 

A straightforward computation shows that   ./1/1 pp AA   First we show that 
    ;[,0] MUUS    

(in particular this proves that   MM  ). One can show that   is convex on 

the subset of all operators in X  having the spectrum contained in the positive 

semiaxis. In particular,   is convex on .M  Direct computations yield 

 

       ./1/1/1/1 pppp AAUAAU    
 

Thus   MU   for all U  with spectrum   [.,0] US  Now we prove that 

MM :  is a contraction, with contraction constant .
1

p

p
q


  Precisely we 

prove that 

  .,
1

MU
p

p
U 


  

We have: 

 

  .,
1

1

0;
1

MU
p

p
UIAUI

AUIAUMUAUI
p

p
U

p

ppp



















 

 



20 

OLTEANU 

Now the conclusion on   being a contraction follows by a standard 

differential calculus argument. Now application of contraction theorem and 

an elementary computation shows that 

 

  .,
1

1

1/1

00

/1
NkAI

p

p
UU

q

q
AU

A

p

A

kk
p

k














 






  

The proof is complete.                                                                                                      
□ 

 

 

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DEPART MENT OF MATHE MAT ICS -IN FOR MAT ICS, UN IVERSITY P OL ITEHN ICA OF 

, 060042 BUCHAREST, ROMANIA