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N.P.Sah / BIBECHANA 9 (2013) 126-129 : BMHSS, p.126 (Online Publication:  Nov., 2012) 
 
 

BIBECHANA 
A Multidisciplinary Journal of Science, Technology and Mathematics 

ISSN 2091-0762 (online) 

Journal homepage: http://nepjol.info/index.php/BIBECHANA 

 

 

About Riesz theory of compact operators 
 

Nagendra Pd. Sah 
Department of Mathematics, M.M.A.M.Campus (Tribhuvan University) Biratnagar, Nepal. 

Article history: Received 2 October, 2012; Accepted 12 November, 2012 

 
 

Abstract 
In this paper, it is shown that every compact operators are bounded and continuous. The bounded and continuous 

properties of an operator is sufficient for a Riesz operator. For mapping T: K- Iλ  in normed linear space with some 

extended [1] properties, T becomes compact.  

 

Keywords: Chain length; Riesz operators;  Relatively regular operator 

 

1. Introduction 

 

For the mapping T = K- Iλ , in normed linear space, [where I is the identity map, K is compact map and 

0≠λ is an arbitrary scalar from φ ], the fredholm alternative theorem is valid. The null spaces and image 

spaces of T
k
, where k is natural number, form chains with the same chain length. The image space B(T) is 

always closed; K- Iλ  is open and the eigenvalues of K form a null sequence, provided infinitely many 

such eigenvalues exist. An operator which satisfies these properties for all 0≠λ , is known as a Riesz 

operator. If K is a compact map, then T = K- Iλ  possesses a decomposition into an invertible operator 

R(λ ) and an operator S(λ ) of finite rank. This leads to the decisive definitions. The operators that in the 

quotient algebra of all continuous maps with respect to the ideal of all finite dimensional maps, they are 

right or left invertible respectively, such invertible are known as Atkinson operators. Still another 

important notion which stands together very closely with the above definitions. If for a continuous 

operator T, there exists a continuous operator S with T.S.T. = T, then T is called relatively regular 

operator [2], For a topological characterization of these relatively regular operators, if E is a named 

linear space and )E(RK∈ ; then put I-K=T and N(T
k
)=Nk, B(T

k
)=Bk for k=0,1,2… 

 

2. Some properties of null and image spaces  

 

As null spaces form an increasing sequence {0}=N0⊂ N1 ⊂ N2⊂ N3⊂…. 

1. Nk is closed: This follows from the continuity of T
K
, k = 0,1,2…. 

2. dim N1<∞ : If Nx n ∈ , and ||xn||=1, Nn∈ , then (I-K)xn=0, therefore xn=K. xn; on account of the 

compactness of K, the sequence {k.xn} has a subsequence {k.xn} with  

lim  K. xn  = lim 
knx  = y 

∞→k   ∞→k  

 



N.P.Sah / BIBECHANA 9 (2013) 126-129 : BMHSS, p.127 (Online Publication:  Nov., 2012) 

 

Therefore { 1||x||:Ex =∈ } 1N∩  is compact, and hence N1 is finite dimensional. 

3. dim Nk <∞  :   Since 

T
k
 = (I-K)

k
 = I - 




 −++
−

− − k1K2 K.)1(.....K
2

)1k(k
K.k  

 =  I - Kk, Nk∈ , holds and Kk is compact [3]. From above result it follows that dim 

Nk<∞  for all k = 0,1,2… 

4. If there exists  0k ≥  s.t. Nk = Nk+1, then it follows that Nk = Nk+1, .Nn∈  

Proof: If  2kNx +∈ , then T
k+2

.x = T
k+1

(T.x)= 0, T.x k1K NN =∈ + , Therefore, also T
k
(T.x)=T

k+1
.x=0 i.e., 

1kNx +∈ Hence Nk+1 =  Nk+2 and by complete induction, wet get Nk= Nk+n. 

Definition (1:1):  There exists, either smallest number p = p(T)<∞  satisfying Np=Np+1 in which case we 

say that T fulfills the null chain condition with the null chain length p -  or we always have Nk≠ Nk+1, 

therefore p(T) = ∞ in which case we say that T does not fulfill the null chain condition [4]. 

Theorem (2.1): If T = I-K, K )E(R∈ , then Nk is closed, dim Nk < ∞  for all k = 0,1,2…, and T satisfies 

the null chain condition. We now verify the corresponding statements for image spaces: 

Bk is closed: In case k = 1, we must show that y 1B∈ , y 1B∈ , in other words  

           yn=T.xn 1B∈  and yn→y, then we must show that y 1B∈ , if  

(1)  ||ux||glb k
Nu

k

1

−=δ
∈

, then we choose an Nu k ∈ , s.t. ||xk-uk|| ≤ 2. :kδ we put vk = xk-uk, then 

(2) ||vk||≤ 2. :kδ and yk = T.xk = T (vk+uk) = T.vk we first of all show that the sequence {uk}is bounded 

[5]; otherwise let there exists a subsequence which we again denote by {vk} s.t. ||vk|| ∞→ for k ∞→ . For  

wk = 
||v||

v

k

k . The sequence 

(3) {T.wk} = 








=








||y||

y

||v||

v
.T

k

k

k

k  converges to 0. Since {yk} converges by assumption. Since {wk} is 

a boundend s.t, {K.wk} contains a convergent subsequence {K.
jkw }. Then by (3) it follows that 

jkw = 

(I-K) 
jkw + k. 

jkw  z→ . And also by (3) lim T. 
jkw = T.z = 0, hence 1Nz∈  from which it follows 

that. 

||wk-z|| = z
v

ux

k

kk −
−

 

 =  )z.||v||u(x.
||v||

1
kkk

k

+−  

 ≥  
||v|| k

Kδ  

 ≥  
2

1
  

 

 

 



N.P.Sah / BIBECHANA 9 (2013) 126-129 : BMHSS, p.128 (Online Publication:  Nov., 2012) 

 

Since uk + ||vk||.z 1N∈ . This contradicts that lim 
jkw = z : hence {vk} is bounded. Therefore the 

sequence {K. vk }contains a convergent subsequence {K.
ikv } and the sequence{

ikv }= {T. 
ikv +k. 

ikv }={
ii kk u.Ky + } converges to an element v. It follows that 

iky =
ikv - K

ikv →v - K.v = T.v, 

Hence y = T.v 1B∈ [6]. Then from above equation (3) it follows that Bk is closed also for k>1. 

Theorem (2.2)   

a) If T∈L(X) fulfills the null chain condition with the null chain length p≤m iff Nr mB∩ = {0} for some 

r N∈  holds or:  Nm = Nr+m iff Nr mB∩  = {0}, r N∈  

b) T∈L(X) fulfills the image chain condition with the image chain length q n≤ iff there exists a 

component Cr to Br in X which is contained in Nn. 

or :  Bn = Bn+r iff X = Cr ⊕  Br⊂ Nn + Br = X, r  N∈  [7].  

Theorem (2.3): If K ),E(R∈ then ∞<−β=−α )KI()KI( , K possesses a representation K = R+S 

with RS= SR = 0, R )E(R∈ and S )E(R∈ ; I – K possesses a representation (I-R)-S with (I-R)S = S(I-

R), where I-R possesses a continuous inverse on E. 

Definition (1.2): L φ∈  is called eigenvalue of S ∈L(X), X an arbitrary vector space, if there exists an 

x∈X, x 0≠  s.t. (S - L.I).x = 0, therefore 0)I.LS( >−α , x is then called eigensolution corresponding to 

L; N(S-L.I) is called the eigenspace corresponding to L and )I.LS( −α the multiplicity of L. 

Theorem (2.4): If K∈R(E), then for every 0≠λ , then 

 

a) )IK()IK( λ−β=λ−α  

b) )IK(q)IK(p λ−=λ−  

c) )IK(B λ−  is closed 

d) IK λ− is open 

e) the eigenvalues of K form, provided they are infinite in number, a null sequence. 

Theorem (2.5): If If K∈L(E), then every 0≠λ , then 

 

a) )IK()IK( λ−β=λ−α  

b) )IK(q)IK(p λ−=λ−  

c) IK λ−  is relatively regular. 

d) the eigenvalues of K form a null sequence, provided they are infinite in number. 

 

3. Conclusions 

 

K∈L(E) is called a Riesz- operators if IK λ−  possesses the properties (a) – (e)  of theorem (2.4) for 

each φ∈λ . Again K∈L(E) is a Riesz operator iff for every 0≠λ , it has the properties (a) – (d) of 

theorem (2.5). Thefore, every compact operator is always a Riesz-operator. 

 

 

References  

 
[1] A.E. Taylor: Introduction to Functional Analysis, New-York John Wiley and Sons Inc London (1956). 



N.P.Sah / BIBECHANA 9 (2013) 126-129 : BMHSS, p.129 (Online Publication:  Nov., 2012) 
 

[2]  A.P. Robertson and W. Robertson: Topological Vector Spaces Cambridge University Press (1964). 

[3]   J.L Kelley: General topology, D.Van Nostrand, New York (1955). 

[4]     Conway, John.B: A course on Functional Analysis Springer, Verlang (1985). 

[5]     G. Kothe,  Topological Vector Space, I and II, Springer, Verlang, (1969). 

[6]     M.M. Day: Normed Linear Spaces, Berlin, Springer, Verlang, (1958). 

[7]     R.G. Cooke: Linear operators, Macmillan, London, (1953). 

Further Readings 

[8]    A.G. Ramm, American Mathematical Monthly Journal, (2001) 81. 

[9]    T. Kato: Perturbation theory for linear operators, Springer, Verlang (1966).