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1 ISSN: X (Print) ISSN: (Online) Bulletin of the Iranian Mathematical Society Vol 4 (05), No 6, pp 5 57 Title: The Libera operator on Dirichlet spaces Author(s): G Bao and J Yang Published by Iranian Mathematical Society
2 Bull Iranian Math Soc Vol 4 (05), No 6, pp 5 57 Online ISSN: THE LIBERA OPERATOR ON DIRICHLET SPACES G BAO AND J YANG (Communicated by Hamid Reza Ebrahimi Vishki) Abstract In this paper, we consider the boundedness of the Libera operator on Dirichlet spaces in terms of the Schur test Moreover, we get its point spectrum and norm Keywords: Libera operator, Dirichlet spaces, point spectrum, norm MSC(00): Primary: 30H0; Secondary: 47A30 Introduction Let D be the unit disk in the complex plane C Denote by D α, α R, the Hilbert space of all analytic functions f(z) n0 a nz n in D with f(0) 0 and f D α n α a n < n Note that the classical Dirichlet space D is obtained for α 0 See [,,7,9,] for D α spaces The Libera operator L on D α, α <, is defined by Lf(z) z z f(w)dw 0 f(w)dw for all f D α See [4, 6, 0] for the Libera operator on other spaces Rhaly [8] gave a series of results about the Libera operator on the classical Dirichlet space D In this paper, we consider the Libera operator on D α spaces By [], if α > 0, then there exists a constant C such that f(z) C f Dα ( z ) α/ for all f D α Clearly, D α D β when α β Consequently, the Libera operator L is well defined on D α, α < Article electronically published on December 5, 05 Received: March 04, Accepted: October 04 Corresponding author 5 c 05 Iranian Mathematical Society
3 The Libera operator on Dirichlet spaces 5 Let e n (z) n α z n, n,, Then the set {e n } n forms an orthonormal basis for D α Note that Le n (z) z z n α w n dw n α n + (zn + z n + + z) n α n + 0 n α w n dw (n α en (z) + (n ) α en (z) + + e (z) Thus, L has matrix entries 0 i > j, a ij Le j, e i ( ) α j i If f(z) n f(n)e n(z), then Lf(z) j+ n mn i j f(m) ( n ) α e n (z) m + m For α >, a direct calculation gives that f D α is comparable with () f (z) ( z ) α dxdy D Pavlović [6] investigated the Libera operator on mixed norm spaces In particular, under the seminorm () of D α spaces, he showed that if α >, then L is bounded on D α if and only if α < In this paper, by different technique, we give that L is bounded on D α if and only if α < Particularly, we obtain that L is also bounded on D α for α Furthermore, the point spectrum and norm of L on the D α space are also considered Main results Let T be a bounded linear operator on a Hilbert space H Recall that the point spectrum σ p (T ) of T is σ p (T ) {λ C : ker(λi T ) 0}, where I is the identical operator of H The point spectrum of L on the Dirichlet space D was obtained in [8] Using the Schur test, we get the following result Theorem The Libera operator L is bounded on D α if and only if α < Moreover, σ p (L) {λ C : λ α < α } { : n, 3, } n )
4 53 Bao and Yang Proof Let p i i/ i+, i,, Then j i It follows that and a ij p i (j + )j α (j + )j α (j + )j α i i i j i j 0 i ( α)/ i + i max{t α/, (t + ) α/ }dt max{t α/, (t + ) α/ }dt a ij p i α p j, 0 α <, i On the other hand, a ij p j Hence, and j j a ij p i α α p j, α < 0 ji i α i α j + ji j i ( ) α i j / j j + j+ max{t α/, (t/) α/ } dt t max{t α/, (t/) α/ } dt t a ij p j α i / 4 α p i, 0 α <, α α a ijp j α i / pi, α < 0 α j By the Schur test (see [3, P 4]), we get that L is bounded on D α for α < For α, let z n f α(z) n n α log(n + ), that is, f α (z) e n (z) n log(n + ) Then f α D α n n n log (n + ) <
5 The Libera operator on Dirichlet spaces 54 Note that α We deduce that Lf α ( n D α n mn m(m + ) log(m + ) m ( m m(m + ) log(m + ) m +, (m + ) log(m + ) m ) α ) α which implies that L is not bounded on D α, α Thus, L is bounded on D α if and only if α < We now compute the point spectrum of L If f(z) n f(n)e n(z), then Thus, Consequently, which gives If Lf λf, then Lf(z) Lf(n) n α n mn mn f(m) ( n ) α e n (z) m + m f(m) (m + )m α, n,, n α α f(n) Lf(n) (n + ) Lf(n + ), (n + )n α f(n) (n + )Lf(n) n α (n + ) α+ Lf(n + ) λn α (n + ) α+ f(n + ) [λ(n + ) ]f(n) If λ 0, then f(n) 0 for all n Hence, 0 / σ p (L) It follows that Thus, ( n + f(n + ) n f(n) n α n j ) α [ ] f(n) λ(n + ) ( λj ) f(), n We now want to know what nonzero values of λ will result in the convergence of n f(n) Clearly, { n : n, 3, } σ p(l) Suppose that λ / { n :
6 55 Bao and Yang n, 3, } Then Hence, f(n) f(n + ) ( (n + ) n α (n + ) 3 α) λ + (n + ) α (λ + λ) (n + ) α (n + ) 3 α λ (n + ) α (λ + λ) + (n + ) α [ ] f(n) lim n n f(n + ) (α ) λ + (λ + λ) λ By Raabe s test [5, Theorem II, P 396], n f(n) converges for λ+λ λ + α > and diverges for λ+λ λ + α < Namely, the series converges for λ α < α and diverges for λ α > α lim ln n{ n [ f(n) n f(n + ) ] } If λ α α, then ln n { n [ (n + ) n α (n + ) 4 α + ( α)(n + ) 3 α] λ lim n n (n + ) α (n + α ) λ + (n + ) α n(n + ) α } (n + ) α (n + α ) λ + (n + ) α 0 α 3α+ λ λ 0 By [5, Theorem III, P 396], we obtain that the series n f(n) diverges if λ α α Therefore, σ p (L) {λ C : λ α < α } { : n, 3, } n For α <, let L Dα be the norm of L on D α spaces H Rhaly [8] proved that L D Now we give the following result Theorem Let 0 α < Then L Dα α
7 The Libera operator on Dirichlet spaces 56 Proof Let p i i /, q j j/ j+, i, j,, Note that 0 α < We have j a ij p i i α/ i (j + )j α i j i t α/ dt (j + )j α i Similarly, j a ij q j i α i α ji i (j + )j α i j 0 t α/ dt α j + α qj j / j α/ (j + ) i α ji j+ j t α/ dt t α/ dt α i / α pi By the Schur test, we get that L Dα α Here we give the details for the completeness If f(z) n f(n)e n(z), then Lf D α f(m)a nm n m ( ) ( ) f(m) a nm a nmq m q m n α m n m m f(m) a nm q m p n f(m) a nm p n α q m m n ( ) f(m) α m ( ) f D α α Hence L Dα α Since L D α λ for λ σ p (L), we get that L Dα The proof is complete α } If α <, then Theo- Remark 3 Observe that L Dα max{/, rem is not true α
8 57 Bao and Yang Acknowledgments The authors would like to thank the referee for his/her helpful comments G Bao is supported in part by NSF of China (No 3734) J Yang is supported by Innovation Program of Shanghai Municipal Education Commission (No 3YZ090) and NSF of China (No 50357) References [] G Bao and H Wulan, Hankel matrices acting on Dirichlet spaces, J Math Anal Appl 409 (04), no, 8 35 [] L Brown and A Shields, Cyclic vectors in the Dirichlet space, Trans Amer Math Soc 85 (984), no, [3] P Halmos, A Hilbert Space Problem Book, Graduate Texts in Mathematics, 9, Encyclopedia of Mathematics and its Applications, 7, Springer-Verlag, New York-Berlin, 98 [4] M Nowak and M Pavlović, On the Libera operator, J Math Anal Appl 370 (00), no, [5] J Olmsted, Advanced Calculus, Appleton-Century-Crofts, New York, 96 [6] M Pavlović, Definition and properties of the Libera operator on mixed norm spaces, Sci World J 04 (04), Article ID , 5 pages [7] A Persson, On the spectrum of the Cesáro operator on spaces of analytic functions, J Math Anal Appl 340 (008), no, [8] H Rhaly, An averaging operator on the Dirichlet space, J Math Anal Appl 98 (984), no, [9] R Rochberg and Z Wu, A new characterization of Dirichlet type spaces and applications, Illinois J Math 37 (993), no, 0 [0] A Siskakis, Semigroups of composition operators in Bergman spaces, Bull Austral Math Soc 35 (987), no 3, [] D Stegenga, Multipliers of the Dirichlet space, Illinois J Math 4 (980), no, 3 39 (Guanlong Bao) Department of Mathematics, Shantou University, Shantou 55063, China address: glbaoah@63com (Jun Yang) Department of Mathematics, Shanghai Maritime University, Shanghai 0306, China address: yangjundlut@gmailcom
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