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

## Transkript

(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 D2 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 D2 D3 ) O (On ( K D1 K D2 K D3 )) ii) On ( K D K D K D ) O(On ( K D D 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 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] 179 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 ) 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 D2 D3 contains the real closed interval J (see Figure 1). For k =1,2,3, let Jk wDk D1 D2 D3 . 181 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) , 183 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) E 2 ( D) f 2 (1 d i d 3) and V2 ( f1 , f 2 , f 3 ) H ( D) 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. 185 On the Eigenvalues of Integral Operators 186