On the Eigenvalues of Integral Operators


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1 Çanaya Üniversitesi FenEdebiyat 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 simplyconnected 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 halfplanes 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 pth 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 squaresummable 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 halfplane 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 squareroot of T We call H the domain space of S REMARK Suppose that,, are simplyconnected 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 squareroot 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 squareroot 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 squareroot 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 nontangential 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 halfplane, 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 halfplanes, 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 FejerRiesz Type, Çanaya University, Journal of Arts and Sciences, Issue: 5, May 006,
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