Öz
We characterize continuity and compactness of the Volterra integral operator $T_g$ with the non-constant analytic symbol $g$ between certain weighted Fréchet or (LB)-spaces of analytic functions on the open unit disc, which arise as projective (resp. inductive) limits of intersections (resp. unions) of Bergman spaces of order $1<p<\infty$ induced by the standard radial weight $(1-|z|^2)^\alpha$ for $0<\alpha<\infty$. Motivated from the earlier results obtained for weighted Bergman spaces of standard weight, we also establish several results concerning the spectrum of the Volterra operators acting on the weighted Bergman Fréchet space $A^p_{\alpha+}$, and acting on the weighted Bergman (LB)-space $A^p_{\alpha-}$.
Anahtar Kelimeler
Volterra operator weighted spaces of analytic functions Bergman spaces Fréchet spaces (LB)-spaces spectrum
Destekleyen Kurum
TÜBİTAK
Proje Numarası
1059B191800828
Teşekkür
This article was completed during the autor's stay at Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, funded by The Scientific and Technological Research Council of Turkey (TÜBİTAK) with grant number 1059B191800828. The author is deeply thankful to Prof. José Bonet, Prof. Enrique Jordá, and Prof. David Jornet for useful suggestions and kind hospitality.
Kaynakça
- [1] A. Albanese, J. Bonet and W.J. Ricker, The Cesàro operator in growth Banach spaces of analytic functions, Integr. Equ. Oper. Theory 86, 97–112, 2016.
- [2] A. Albanese, J. Bonet and W. Ricker, The Cesàro operator in the Fréchet spaces $\ell^{p+}$ and $L^{p-}$, Glasg. Math. J. 59 (2), 273–287, 2017.
- [3] A. Albanese, J. Bonet and W. Ricker, The Cesàro operator on Korenblum type spaces of analytic functions, Collect. Math. 69 (2), 263–281, 2018.
- [4] A. Aleman and J.A. Cima, An integral operator on $H^p$ and Hardy’s inequality, J. Anal. Math. 85, 157–176, 2001.
- [5] A. Aleman and O. Constantin, Spectra of integration operators on weighted Bergman spaces, J. Anal. Math. 109, 199–231, 2009.
- [6] A. Aleman and J.A. Peláez, Spectra of integration operators and weighted square functions, Indiana Univ. Math. J. 61, 1–19, 2012.
- [7] A. Aleman and A. Persson, Resolvent estimates and decomposable extensions of generalized Cesàro operators, J. Funct. Anal. 258, 67–98, 2010.
- [8] A. Aleman and A.G. Siskakis, An integral operator on $H^p$, Complex Var. Theory Appl. 28, 149–158, 1995.
- [9] A. Aleman and A.G. Siskakis, Integration operators on Bergman spaces, Indiana Univ. Math. J. 46, 337–356, 1997.
- [10] M. Basallote, M.D. Contreras, C. Hernández-Mancera, M.J. Martín and P.J. Paúl,Volterra operators and semigroups in weighted Banach spaces of analytic functions, Collect. Math. 65, 233–249, 2014.
- [11] J. Bonet, The spectrum of Volterra operators on weighted spaces of entire functions, Quart. J. Math. 66, 799–807, 2015.
- [12] J. Bonet, The spectrum of Volterra operators on Korenblum type spaces of analytic functions, Integr. Equ. Oper. Theory 91, 46, 2019.
- [13] Ž. Čučković and R. Zhao, Weighted composition operators between different weighted Bergman spaces and different Hardy spaces, Ilinois J. Math. 51 (2), 479–498, 2007.
- [14] P. Duren, Theory of $H^p$ Spaces, Academic Press, New York, 1970.
- [15] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman Spaces, Graduate Texts in Math. 199, Springer, New York, 2000.
- [16] H. Jarchow, Locally Convex Spaces, Teubner, Stuttgart, 1981.
- [17] E. Kızgut, The Cesàro operator on weighted Bergman Fréchet and (LB)-spaces of analytic functions, Filomat, to appear.
- [18] R. Korhonen and J. Rättyä, Intersections and unions of weighted Bergman spaces, Comput. Methods Funct. Theory 5 (2), 459–469, 2005.
- [19] B. Malman, Spectra of generalized Cesàro operators acting on growth spaces, Integr. Equ. Oper. Theory 90, 26, 2018.
- [20] R. Meise and D. Vogt, Introduction to Functional Analysis, Oxford Graduate Texts in Mathematics, Clarendon Press, Oxford, 1997.
- [21] C. Pommerenke, Schlichte Funktionen un analytische Funktionen von beschränkter mittlerer Oszilation, Comment. Math. Helv. 52, 591–602, 1977.
- [22] J. Rättyä, Integration operator acting on Hardy and weighted Bergman spaces, Bull. Aust. Math. Soc. 75, 431–446, 2006.
- [23] A. Siskakis, Volterra operators on spaces of analytic functions-a survey, in: Proceedings of the First Advanced Course in Operator Theory and Complex Analysis, 51–68, Univ. Sevilla Serc. Publ., Seville, 2006.
- [24] K. Zhu, Operator Theory on Function Spaces, Mathematical surveys and monographs 138, American Mathematical Society, 2007.
Öz
Proje Numarası
1059B191800828
Kaynakça
- [1] A. Albanese, J. Bonet and W.J. Ricker, The Cesàro operator in growth Banach spaces of analytic functions, Integr. Equ. Oper. Theory 86, 97–112, 2016.
- [2] A. Albanese, J. Bonet and W. Ricker, The Cesàro operator in the Fréchet spaces $\ell^{p+}$ and $L^{p-}$, Glasg. Math. J. 59 (2), 273–287, 2017.
- [3] A. Albanese, J. Bonet and W. Ricker, The Cesàro operator on Korenblum type spaces of analytic functions, Collect. Math. 69 (2), 263–281, 2018.
- [4] A. Aleman and J.A. Cima, An integral operator on $H^p$ and Hardy’s inequality, J. Anal. Math. 85, 157–176, 2001.
- [5] A. Aleman and O. Constantin, Spectra of integration operators on weighted Bergman spaces, J. Anal. Math. 109, 199–231, 2009.
- [6] A. Aleman and J.A. Peláez, Spectra of integration operators and weighted square functions, Indiana Univ. Math. J. 61, 1–19, 2012.
- [7] A. Aleman and A. Persson, Resolvent estimates and decomposable extensions of generalized Cesàro operators, J. Funct. Anal. 258, 67–98, 2010.
- [8] A. Aleman and A.G. Siskakis, An integral operator on $H^p$, Complex Var. Theory Appl. 28, 149–158, 1995.
- [9] A. Aleman and A.G. Siskakis, Integration operators on Bergman spaces, Indiana Univ. Math. J. 46, 337–356, 1997.
- [10] M. Basallote, M.D. Contreras, C. Hernández-Mancera, M.J. Martín and P.J. Paúl,Volterra operators and semigroups in weighted Banach spaces of analytic functions, Collect. Math. 65, 233–249, 2014.
- [11] J. Bonet, The spectrum of Volterra operators on weighted spaces of entire functions, Quart. J. Math. 66, 799–807, 2015.
- [12] J. Bonet, The spectrum of Volterra operators on Korenblum type spaces of analytic functions, Integr. Equ. Oper. Theory 91, 46, 2019.
- [13] Ž. Čučković and R. Zhao, Weighted composition operators between different weighted Bergman spaces and different Hardy spaces, Ilinois J. Math. 51 (2), 479–498, 2007.
- [14] P. Duren, Theory of $H^p$ Spaces, Academic Press, New York, 1970.
- [15] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman Spaces, Graduate Texts in Math. 199, Springer, New York, 2000.
- [16] H. Jarchow, Locally Convex Spaces, Teubner, Stuttgart, 1981.
- [17] E. Kızgut, The Cesàro operator on weighted Bergman Fréchet and (LB)-spaces of analytic functions, Filomat, to appear.
- [18] R. Korhonen and J. Rättyä, Intersections and unions of weighted Bergman spaces, Comput. Methods Funct. Theory 5 (2), 459–469, 2005.
- [19] B. Malman, Spectra of generalized Cesàro operators acting on growth spaces, Integr. Equ. Oper. Theory 90, 26, 2018.
- [20] R. Meise and D. Vogt, Introduction to Functional Analysis, Oxford Graduate Texts in Mathematics, Clarendon Press, Oxford, 1997.
- [21] C. Pommerenke, Schlichte Funktionen un analytische Funktionen von beschränkter mittlerer Oszilation, Comment. Math. Helv. 52, 591–602, 1977.
- [22] J. Rättyä, Integration operator acting on Hardy and weighted Bergman spaces, Bull. Aust. Math. Soc. 75, 431–446, 2006.
- [23] A. Siskakis, Volterra operators on spaces of analytic functions-a survey, in: Proceedings of the First Advanced Course in Operator Theory and Complex Analysis, 51–68, Univ. Sevilla Serc. Publ., Seville, 2006.
- [24] K. Zhu, Operator Theory on Function Spaces, Mathematical surveys and monographs 138, American Mathematical Society, 2007.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Matematik |
| Bölüm | Matematik |
| Yazarlar | |
| Proje Numarası | 1059B191800828 |
| Yayımlanma Tarihi | 6 Ağustos 2021 |
| Yayımlandığı Sayı | Yıl 2021 |