(12) Y ksel Soykan - Journal Of Arts and Sciences

Çankaya Üniversitesi Fen-Edebiyat Fakültesi,
Journal of Arts and Sciences Say›: 6 / Aral›k 2006
On the Eigenvalues of Integral Operators
Yüksel SOYKAN1
Abstract
In this paper, we obtain asymptotic estimates of the eigenvalues of certain positive integral operators.
Key words: Positive Integral Operators, Eigenvalues, Hardy Spaces.
Özet
Bu çal›flmada baz› positive integral operatörlerin özde¤erlerinin asimtotik yaklafl›mlar›n› elde edece¤iz.
Anahtar Kelimeler: Pozitif ‹ntegral Operatörleri, Özde¤erler, Hardy Uzaylar›.
1. INTRODUCTION
From now on, let J be a fixed closed subinterval of the real line R. Suppose that D
is a simply-connected domain containing the real closed interval J and ϕ is any
function, which maps D conformally onto ∆, where ∆ is the open unit disk of complex
plane C. Let us define a function KD on D x D by
1
2
1
2
M c(] ) M c( z )
1 M (] )M ( z )
K D (] , z )
for all ] , z  D,
1
for either of the branches of M c 2 . The function KD is independent of the choice of mapping
function ϕ, see [1, p.410]. By restricting the function KD to the square JxJ we obtain a
compact symmetric operator TD on L2 defined by
––––––––––––––––––––––––––––––
1 Karaelmas Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Zonguldak.
e-mail: [email protected]
177
On the Eigenvalues of Integral Operators
( f  L2 ( J ), s  J ).
This operator is always positive in the sense of operator theory (i.e. ¢Tf , f ² t 0
2
for all f  L ( J ) , see [1].
³
TD f ( s )
We shall use
J
K D ( s, t ) f (t )dt
On ( K D ) to denote the eigenvalues of TD .
In this work the following theorem shall be proved in detail.
THEOREM 1.1 If D1 , D2 , D3 are three half-planes and their boundary lines are
D1 ˆ D2 ˆ D3 contains the real closed interval J,
not parallel pairwise and if D
then
On ( K D ) ≅ On ( K D K D K D )
where an ≅ bn means an
1
2
3
O(bn ) and bn
O(an ) .
To prove Theorem 1.1 we will show that
i) On ( K D1 ˆ D2 ˆ D3 ) O (On ( K D1 K D2 K D3 ))
ii)
On ( K D K D K D ) O(On ( K D ˆ D ˆ D )).
1
2
3
1
2
3
This is a special case of a theorem in [1, Theorem 1] and we give a different proof.
2. PRELIMINARIES
f
The space H (' ) is just the set of all bounded analytic function on ' with the
p
uniform norm. For 1 d p f , H (' ) is the set of all functions f analytic on ' such
that
1
0 r 1 2S
sup
³
2S
0
p
f (reiT ) dT f.
(1)
p
The p-th root of the left hand side of (1) here defines a complete norm on H (' ) .
2
For more information on this spaces see [2 and 3 ]. In the case of p=2, H be the
familiar Hardy space of all functions analytic on ' with square-summable Maclaurin
coefficients.
C
‰ f and let M be a Riemann
Let D be a simply connected domain in C
mapping function for D, that is, a conformal map of D onto ' . An analytic function g
2
2
on D is said to be of class E ( D ) if there exists a function f  H (' ) such that
178
Yüksel SOYKAN
1
1
f M ( z ) M c( z ) 2 ( z  D) where M c 2 is a branch of the square root of M c . We
define g E 2 ( D )
f H 2 ( ' ) . Thus, by construction, E 2 ( D) is a Hilbert space with
g ( z)
¢ g1 , g 2 ² E 2 ( D )
where gi ( z )
¢ f1 , f 2 ² H 2 ( ' )
1
f i M ( z ) M c( z ) 2 , (i = 1, 2) and the map UM : H 2 (') o E 2 ( D)
given by
UM f ( z )
1
f (M ( z ))M c( z ) 2 ( f  H 2 ('), z  D)
is an isometric bijection. For more information on this spaces see [1]. If wD is a
rectifiable Jordan curve then the same formula
VM f ( z )
1
f (M ( z ))M c( z ) 2 ( f  L2 (w'), z  wD)
2
2
2
defines an isometric bijection VM of L (w' ) onto L (wD ) , the L
space of
normalized arc length measure on wD where wD and w' denote the boundary of D
and ' respectively. The inverse
VM 1 : L2 (wD) o L2 (w' )
V\
of V\ is given by
V\ g ( w)
g (\ ( w))\ c w) ( g  L2 (wD), w  w',\
M 1 ).
To prove Theorem 1.1 we need the following lemma. This is Corollary 1.3 to
Lemma 1.2 in [4].
LEMMA 2.1 Suppose that D is a disc or a codisc or a half-plane and
J c Ž D be a
2
circular arc (or a straight line) then for every g  E ( D),
1
2S
³
J
2
g (z ) dz d g
1
2S
2
E2 ( D)
c
³
2
g (z ) dz .
wD
Suppose now that D contains our fixed interval J. By restricting M to J we obtain a
2
2
linear operator S D : E ( D ) o L ( J ) defined by S D f ( s ) f ( s ) (f  E 2 ( D ), s  J ) .
*
Then S D is compact operator and TD S D S D is the compact, positive integral
operator on J with kernel K D :
1
2
1
2
M c( s) M c(t )
K D ( s, t )
1 M ( s )M (t )
for all s,t ∈ J. This is proved in [1]
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On the Eigenvalues of Integral Operators
DEFINITION 2.1 Let H and H c be Hilbert spaces and suppose that T is a
compact, positive operator on H. If S : H c o H is a compact operator such that
T SS * , then S is called a quasi square-root of T. We call H c the domain space of S.
REMARK 2.2 Suppose that D1 , D2 , D3 are simply-connected domains containing
2
J and let TD1 , TD2 , TD3 be continuous positive operators on a Hilbert space L ( J ) and
2
suppose that for each i, S Di is a quasi square-root of TDi with domain space E ( Di ) .
3
T ( K D1 K D2 K D3 ) T (¦ K Di )
If T
i 1
( s, t ) K D2 ( s, t ) K D3 ( s, t )) f (t )dt ( f  L2 ( J ), s  J ), then
T : L ( J ) o L ( J ) is compact, positive integral operator and T has the quasi
so
that
2
T f ( s)
2
³ (K
J
D1
square-root
S : E 2 ( D1 ) E 2 ( D2 ) E 2 ( D3 ) o L2 ( J ),
S ( f1 f 2 f 3 )
S D1 f1 S D2 f 2 S D3 f 3
so that
S ( f1 f 2 f3 )( s )
LEMMA 2.3
S D1 f1 ( s ) S D2 f 2 ( s) S D3 f3 ( s )
f1 ( s) f 2 ( s) f3 ( s)
( f  L2 ( J ), s  J ).
Let T1 , T2 be compact operators on a Hilbert space H and suppose
that S1 , S 2 are quasi square-root of T1 , T2 with domain H1 , H 2 respectively.
i) If there exists a continuous operator V : H 2 o H1 such that S 2
VS1 then
(T2 f , f ) d k (T1 f , f ) for some k>0 and so On (T2 ) O(On (T1 )) (n t 0).
ii) If there exists continuous operators V : H 2 o H1 and W : H o H such that
S 2 WS1V , then On (T2 ) O(On (T1 )) .
Proof. See [1, page 407].
3. PROOF OF MAIN RESULT
Suppose that D is a simply connected and bounded domain. Let M be a Riemann
1
mapping function for D and suppose that \ M is the inverse function of M . An
f
analytic function f on D is said to be of class H ( D ) if it is bounded on D.
PROPOSITION 3.1
180
1
f
2
If \ c  H , then H ( D ) Ž E ( D ) .
Yüksel SOYKAN
f
Proof. Suppose that f  H ( D ) . For z  D , define g ( z )
f
Then f $\  H ( D ) and
1
2
1
f \ ( z ) \ c( z ) 2 .
1
2
\ c  H 2 . It follows that ( f $\ )\ c  H 2 . Hence
f  E 2 ( D) .
PROPOSITION 3.2
i) \ c  H
Suppose that wD is a rectifiable Jordan curve, then
1
ii) Each function f  E 2 ( D) has a non-tangential limit f  L2 (wD ) . The map
f o f is an isometric isomorphism and f
2
E2 ( D)
1
2S
³
2
f (z ) dz .
wD
iii) If D is a convex region, it is a Smirnov domain.
iv) If D is a Smirnov domain, then polynomials (thus H f ( D) ) are dense in E 2 ( D) .
2
2
v) E ( D ) coincides with the L (wD ) closure of the polinomials if and only if D
is a Smirnov domain.
Proof. See [2, pages 44, 170 and 173]. For the definition of Smirnov domain see [2,
page 173].
LEMMA 3.4 If D is a disc, or codisc or half-plane, then the formula
Pf ( z )
1
f (] d] x
³
2S i w D ] z
2
2
defines a continuous linear operator P : L (wD ) o E ( D ) with P
1.
Proof. See [1, page 423].
From now on, suppose now that D1 , D2 , D3 are three half-planes, and let
D
D1 ˆ D2 ˆ D3 contains the real closed interval J (see Figure 1). For k =1,2,3, let
Jk
wDk ˆ D1 ˆ D2 ˆ D3 .
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On the Eigenvalues of Integral Operators
D3
D1
J1
D
J3
J
J2
D2
Figure 1.
We shall exhibit continuous operators
N : E 2 ( D) o E 2 ( D1 ) † E 2 ( D2 ) † E 2 ( D3 ) and M : E 2 ( D1 ) † E 2 ( D2 ) † E 2 ( D3 ) o E 2 ( D).
2
To define N suppose first that G={f: f is a polynomial in E ( D ) }. Since D is
convex, G
E 2 ( D) . If f  G , then for all z  D , Cauchy's Integral Formula gives
f ( z)
1
f(w
1
f(w
1
f(w
dw dw dw
³
³
³
2S i J1 w z
2S i J 2 w z
2S i J 3 w z
For f  G and 1 d k d 3 , define a function f k on Dk by
fk ( z)
1
f(w
dw
³
2S i J k w z
( z  Dk ) ,
and define a function fk on wDk by
fk ( z )
182
­ f (] ), if ]  J k
®
¯if ]  w Dk J k ( z  w Dk ).
Yüksel SOYKAN
LEMMA 3.5
If f  G and 1 d k d 3 then
2
2
i) fk  L (wDk ) and f k  E ( Dk ) .
ii) The formula V1 f
( f1 , f 2 , f3 ) defines a continuous linear operator
V1 : G o E 2 ( D1 ) † E 2 ( D2 ) † E 2 ( D3 ) and so that V1 has an extension N by
2
continuity to E ( D ) .
Proof. i) Let
Mk be a Riemann mapping function for Dk and suppose VM and
k
U M k are as in Section 2. The map Pk : L2 (wDk ) o E 2 ( Dk ) given by
1
f (] d]
³
2S i w Dk ] z
Pk f ( z )
is a continuous linear operator with Pk
1 (from Lemma 3.4). Since f k
2
2
2
Pk fk , it
2
follows that f k  E ( Dk ) . So ( f1 , f 2 , f 3 )  E ( D1 ) † E ( D2 ) † E ( D3 ) . Since
fk
2
E 2 ( Dk )
Pk fk
and
V1 f
2
2
E ( Dk )
2
E 2 ( D1 ) † E 2 ( D2 ) † E 2 ( D3 )
2
L ( wDk )
( f1 , f 2 , f3 )
f1
1
2S
2
d fk
2
E 2 ( D1 )
³J
2
f (] ) d ] d f
k
2
E2 ( D)
2
E 2 ( D1 ) † E 2 ( D2 ) † E 2 ( D3 )
f2
2
E 2 ( D2 )
f2
2
E 2 ( D3 )
d3 f
2
E2 ( D)
f o ( f1 , f 2 , f3 ) is a continuous linear operator
G o E ( D1 ) † E ( D2 ) † E ( D3 ) . Now suppose that N is an extension by
2
2
continuity to E ( D ) . Note that then N d 3 .
it follows that the map
2
2
2
If we denote F
H f ( D1 ) † H f ( D2 ) † H f ( D3 ) then F
LEMMA 3.6
The map V2 : F o E ( D ) , is given by
V2 ( f1 , f 2 , f3 )( z )
E 2 ( D1 ) † E 2 ( D2 ) † E 2 ( D3 ) .
2
f1 ( z ) f 2 ( z ) f3 ( z ) ,
(( f1 , f 2 , f 3 )  F , z  D) ,
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On the Eigenvalues of Integral Operators
is a continuous operator so that V2 has an extension M by continuity to
E 2 ( D1 ) † E 2 ( D2 ) † E 2 ( D3 ) .
Proof. If ( f1 , f 2 , f 3 )  F then by Propositions 3.1 and 3.2, fi  H f ( D) Ž E 2 ( D)
f
2
(1 d i d 3) and V2 ( f1 , f 2 , f 3 )  H ( D) Ž E ( D ) . So we have
V2 ( f1 , f 2 , f3 ))
2
f1 f 2 f3
E2 ( D)
2
= f1
E2 ( D)
2
E2 ( D)
2
f2
E2 ( D)
f3
2
E2 ( D)
2 Re al¢ f1 , f 2 ² E 2 ( D )
2 Re al¢ f1 , f3 ² E 2 ( D ) 2 Re al¢ f 2 , f3 ² E 2 ( D )
d f1
2
E2 ( D)
3( f1
2
f2
2
E2 ( D)
E2 ( D)
( f1
2
f2
2
f3
E2 ( D)
E2 ( D)
2
E2 ( D)
f3
2
f3
2
( f1
E2 ( D)
) ( f2
E2 ( D)
E2 ( D)
2
f2
2
E2 ( D)
2
E2 ( D)
f3
)
2
E2 ( D)
)
)
(by Lemma 2.1)
d 3(3 f1
2
E 2 ( D1 )
3 f2
9 ( f1 , f 2 , f3 )
2
E 2 ( D2 )
3 f3
2
E 2 ( D3 )
)
2
E 2 ( D1 ) † E 2 ( D2 ) † E 2 ( D3 )
2
Hence V2 d 9 and V2 is a continuous linear operator. Let now M be extension
2
2
2
2
by continuity to E ( D1 ) † E ( D2 ) † E ( D3 ) . Note that then M d 9 .
PROOF OF THEOREM 1.1
i) Suppose that V1 is as in Lemma 3.5. Note that here T+
S+ S +* and
S D S D* . By definition of V1 , we have S D f S V1 f for every f  G . Thus, by
2
continuity of V1 , S D f S Nf for every f  E ( D) and so S D S N .
2
So for g  L ( J ) ,
TD
¢ S D S D* g , g ²
S D* g
d N*
2
2
N * S* g
S * g
2
d 3¢ S S * g , g ².
184
2
2
N ¢ S S* g , g ²
Yüksel SOYKAN
*
*
That is, S D S D d 2 S S . Hence by Lemma 2.3
On ( S D S D* ) d 3On ( S S* )
as required.
ii) Suppose that V2 is as in Lemma 3.6. By definition of V2 , it follows that
S + ( f1 , f 2 , f3 )
S DV2 ( f1 , f 2 , f3 ) for every ( f1 , f 2 , f3 )  H f ( D1 ) † H f ( D2 ) † H f ( D3 ) . Thus,
by continuity of V2 ,
S + ( f1 , f 2 , f 3 )
and so S +
S D M ( f1 , f 2 , f 3 ) for every ( f1 , f 2 , f 3 )  E 2 ( D1 ) † E 2 ( D2 ) † E 2 ( D3 )
S D M . So for g  L2 ( J ) , we have
2
¢ S S * g , g ² d M ¢ S D S D* g , g ²
d 9¢ S D S D* g , g ².
*
*
i.e, S S d 9 S D S D . Consequently, from Lemma 2.3,
On ( S S* ) d 9On ( S D S D* ) .
REFERENCES
Little, G., “Equivalences of positive integrals operators with rational kernels”, Proc. London. Math. Soc. (3) 62
(1991), 403-426.
Duren, P. L., “Theory of spaces”, Academic Press, New York, 1970
Koosis, P., “Introduction to spaces”, Cambridge University Press, Cambridge, Second Edition, 1998.
Soykan Y., “An Inequality of Fejer-Riesz Type”, Çankaya University, Journal of Arts and Sciences, Issue: 5, May
2006, 51-60.
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On the Eigenvalues of Integral Operators
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