On the Eigenvalues of Integral Operators

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1 Çanaya Üniversitesi Fen-Edebiyat Faültesi, Journal of Arts and Sciences Say : 6 / Aral 006 On the Eigenvalues of Integral Operators Yüsel SOYKAN 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 asimtoti yalafl mlar n elde edece iz Anahtar Kelimeler: Pozitif ntegral Operatörleri, Özde erler, Hardy Uzaylar INTROUCTION From now on, let J be a fixed closed subinterval of the real line R Suppose that is a simply-connected domain containing the real closed interval J and ϕ is any function, which maps conformally onto, where is the open unit dis of complex plane C Let us define a function K on x by K ( ) ( z) (, z) for all, z, ( ) ( z) for either of the branches of The function K is independent of the choice of mapping function ϕ, see [, p40] By restricting the function K to the square JxJ we obtain a compact symmetric operator T on L defined by Karaelmas Üniversitesi, Fen Edebiyat Faültesi, Matemati Bölümü, Zongulda 77

2 On the Eigenvalues of Integral Operators T f() s K(,) s t f() t dt ( f L ( J), s J) J This operator is always positive in the sense of operator theory (ie Tf, f 0 for all f L ( J), see [] We shall use n( K) to denote the eigenvalues of T In this wor the following theorem shall be proved in detail THEOREM If,, are three half-planes and their boundary lines are not parallel pairwise and if contains the real closed interval J, then n( K) n( K K ) K where an b n means an O( bn) and bn O( an) To prove Theorem we will show that i) n( K ) ( ( )) O n K K K ii) ( K K K ) O( ( K )) n n This is a special case of a theorem in [, Theorem ] and we give a different proof PRELIMINARIES The space H ( ) is just the set of all bounded analytic function on with the p uniform norm For p, H ( ) is the set of all functions f analytic on such that i p sup f( re ) d 0 r 0 () p The p-th root of the left hand side of () here defines a complete norm on H ( ) For more information on this spaces see [ and ] In the case of p=, H be the familiar Hardy space of all functions analytic on with square-summable Maclaurin coefficients Let be a simply connected domain in C C and let be a Riemann mapping function for, that is, a conformal map of onto An analytic function g on is said to be of class E ( ) if there exists a function f H ( ) such that 78

3 Yüsel SOYKAN ( ) ( ) ( ) gz f z z ( z ) where is a branch of the square root of We define g f Thus, by construction, E ( ) is a Hilbert space with E ( ) H ( ) g, g f, f E H where ( ) ( ) ( ) ( ) ( ) gi z fi z z, (i =, ) and the map U : H ( ) E ( ) given by U f( z) f( ( z)) ( z) ( f H ( ), z ) is an isometric bijection For more information on this spaces see [] If rectifiable Jordan curve then the same formula V f( z) f( ( z)) ( z) ( f L ( ), z ) is a defines an isometric bijection V of L ( ) onto L ( ), the L space of normalized arc length measure on where and denote the boundary of and respectively The inverse V V : L ( ) L ( ) of V is given by Vgw g w w g L w ( ) ( ( )) ) ( ( ),, ) To prove Theorem we need the following lemma This is Corollary to Lemma in [4] LEMMA Suppose that is a disc or a codisc or a half-plane and be a circular arc (or a straight line) then for every g E ( ), g (z) dz g g (z) dz E ( ) Suppose now that contains our fixed interval J By restricting to J we obtain a linear operator S : E ( ) L ( J) defined by S f ( s) f ( s) ( f E ( ), s J) Then S is compact operator and T SS is the compact, positive integral operator on J with ernel K : () s () t K (,) s t ( s) ( t) for all s,t J This is proved in [] 79

4 On the Eigenvalues of Integral Operators EFINITION Let H and H be Hilbert spaces and suppose that T is a compact, positive operator on H If S: H H is a compact operator such that T SS, then S is called a quasi square-root of T We call H the domain space of S REMARK Suppose that,, are simply-connected domains containing J and let T, T, T be continuous positive operators on a Hilbert space L ( J ) and suppose that for each i, S i is a quasi square-root of T i with domain space E ( i ) If T T( K K ) ( ) K T K i so i T f( s) ( K ( st, ) K (, ) (, )) ( ) ( ( ), ), J st K st f tdt fl J sj that T : L ( J) L ( J) is compact, positive integral operator and T has the quasi square-root S : E ( ) E ( ) E ( ) L ( J), so that S ( f f f ) S f S f S f then S f f f s S f s S f s S f s f s f s f s f L J sj ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ), ) LEMMA Let T, T be compact operators on a Hilbert space H and suppose that S, S are quasi square-root of T, T with domain H, H respectively i) If there exists a continuous operator V : H H such that S VS then ( Tf, f) Tf (, f) for some >0 and so n( T) O( n( T)) ( n 0) ii) If there exists continuous operators V : H H and W : H H such that S WSV, then n( T) O( n( T)) Proof See [, page 407] PROOF OF MAIN RESULT Suppose that is a simply connected and bounded domain Let be a Riemann mapping function for and suppose that is the inverse function of An analytic function f on is said to be of class H ( ) if it is bounded on PROPOSITION If H, then H ( ) E ( ) 80

5 Proof Suppose that f H ( ) For z, define Yüsel SOYKAN gz ( ) f ( z) ( z) Then f H ( ) and H It follows that ( f ) H f E ( ) Hence f PROPOSITION i) H Suppose that is a rectifiable Jordan curve, then ii) Each function f E ( ) has a non-tangential limit f L ( ) f is an isometric isomorphism and f f (z) dz E ( ) The map iii) If is a convex region, it is a Smirnov domain iv) If is a Smirnov domain, then polynomials (thus H ( ) ) are dense in E ( ) v) E ( ) coincides with the is a Smirnov domain L ( ) closure of the polinomials if and only if Proof See [, pages 44, 70 and 7] For the definition of Smirnov domain see [, page 7] LEMMA 4 If is a disc, or codisc or half-plane, then the formula Pf( z) f ( d i x z defines a continuous linear operator P : L ( ) E ( ) with P Proof See [, page 4] From now on, suppose now that,, are three half-planes, and let contains the real closed interval J (see Figure ) For =,,, let 8

6 On the Eigenvalues of Integral Operators J Figure We shall exhibit continuous operators N : E ( ) E ( ) E ( ) E ( ) and M : E ( ) E ( ) E ( ) E ( ) To define N suppose first that G={f: f is a polynomial in E ( )} Since is convex, G E ( ) If f G, then for all z, Cauchy's Integral Formula gives f(w f(w f(w f ( z) dw dw dw i wz i wz i wz For f G and, define a function f on by f(w f ( z) dw i ( z ), w z and define a function f on by f ( ), if f ( z) if ( z ) 8

7 Yüsel SOYKAN LEMMA 5 If f G and then i) f L ( ) and f E ( ) ii) The formula Vf ( f, f, f) defines a continuous linear operator V : GE ( ) E ( ) E ( ) and so that V has an extension N by continuity to E ( ) Proof i) Let be a Riemann mapping function for and suppose V and are as in Section The map P : L ( ) E ( ) given by U P f( z) f ( d i z is a continuous linear operator with P (from Lemma 4) Since f P f, it follows that f E ( ) So ( f, f, f ) E ( ) E ( ) E ( ) Since and f P f f f( ) d f E ( ) E ( ) E ( ) L ( ) Vf ( f, f, f) E ( ) E ( ) E ( ) E ( ) E ( ) E ( ) f f f f E ( ) E ( ) E ( ) E ( ) it follows that the map f ( f, f, f) is a continuous linear operator GE ( ) E ( ) E ( ) Now suppose that N is an extension by continuity to E ( ) Note that then N If we denote F H ( ) H ( ) H ( ) then F E ( ) E ( ) E ( ) LEMMA 6 The map V : F E ( ), is given by V( f, f, f)( z) f( z) f( z) f( z), (( f, f, f) F, z ), 8

8 On the Eigenvalues of Integral Operators is a continuous operator so that V has an extension M by continuity to E ( ) E ( ) E ( ) Proof If ( f, f, f) F then by Propositions and, fi H ( ) E ( ) ( i ) and V ( f, f, f ) H ( ) E ( ) So we have (,, )) E ( ) E ( ) = f f f Real f, f E ( ) E ( ) E ( ) E ( ) V f f f f f f Real f, f Real f, f E ( ) E ( ) f f E f ( f f ( ) ( ) ( ) ( ) ) E E E E ( ) f f f f E ( ) E ( ) E ( ) E ( ) ( f f f ) E ( ) E ( ) E ( ) ( ) ( ) (by Lemma ) ( ) f f f E ( ) E ( ) E ( ) E ( ) E ( ) E ( ) 9 ( f, f, f ) Hence V 9 and V is a continuous linear operator Let now M be extension by continuity to E ( ) E ( ) E ( ) Note that then M 9 PROOF OF THEOREM i) Suppose that V is as in Lemma 5 Note that here T+ S+ S+ and T SS By definition of V, we have S f SV f for every f G Thus, by continuity of V, S f SNf for every f E ( ) and so S S N So for g L ( J),, S S g g S g N S g N S g N S S g, g SS gg, 84

9 Yüsel SOYKAN That is, S S S S Hence by Lemma as required n( SS ) n( S S ) ii) Suppose that V is as in Lemma 6 By definition of V, it follows that S+ ( f, f, f) SV( f, f, f) for every ( f, f, f) H ( ) H ( ) H ( ) Thus, by continuity of V, S ( f, f, f ) S M( f, f, f ) for every ( f, f, f ) E ( ) E ( ) E ( ) + and so S+ S M So for ie, SS g L ( J), we have SS gg, M SS gg, 9 SS g, g 9 SS Consequently, from Lemma, n( SS ) 9 n( SS ) REFERENCES Little, G, Equivalences of positive integrals operators with rational ernels, Proc London Math Soc () 6 (99), uren, P L, Theory of spaces, Academic Press, New Yor, 970 Koosis, P, Introduction to spaces, Cambridge University Press, Cambridge, Second Edition, 998 Soyan Y, An Inequality of Fejer-Riesz Type, Çanaya University, Journal of Arts and Sciences, Issue: 5, May 006,

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