SOME PROPERTIES OF THE INTEGRATION OPERATORS ON THE SPACES ${F(p,q,s)}$*

doi:10.1007/s10473-024-0109-z

Abstract

Abstract: We study the closed range property and the strict singularity of integration operators acting on the spaces $F(p,p\alpha-2,s)$. We completely characterize the closed range property of the Volterra companion operator $I_g$ on $F(p,p\alpha-2,s)$, which generalizes the existing results and answers a question raised in [A. Anderson, Integral Equations Operator Theory, 69 (2011), no. 1, 87-99]. For the Volterra operator $J_g$, we show that, for $0<\alpha\leq1$, $J_g$ never has a closed range on $F(p,p\alpha-2,s)$. We then prove that the notions of compactness, weak compactness and strict singularity coincide in the case of $J_g$ acting on $F(p,p-2,s)$.

Key words: integration operator, closed range property, strict singularity

CLC Number:

47G10

Jiale Chen. SOME PROPERTIES OF THE INTEGRATION OPERATORS ON THE SPACES ${F(p,q,s)}$^*[J].Acta mathematica scientia,Series B, 2024, 44(1): 173-188.

Trendmd

References

[1] Aleman A, Cima J A. An integral operator on $H^p$ and Hardy's inequality. J Anal Math, 2001, 85: 157-176
[2] Aleman A, Siskakis A G. An integral operator on $H^p$. Complex Variables Theory Appl, 1995, 28(2): 149-158
[3] Aleman A, Siskakis A G. Integration operators on Bergman spaces. Indiana Univ Math J, 1997, 46(2): 337-356
[4] Anderson A. Some closed range integral operators on spaces of analytic functions. Integral Equations Operator Theory, 2011, 69(1): 87-99
[5] Anderson A, Jovovic M, Smith W. Some integral operators acting on $H^{\infty}$. Integral Equations Operator Theory, 2014, 80(2): 275-291
[6] Blasco O. Operators on weighted Bergman spaces ($0<p\leq1$) andapplications. Duke Math J, 1992, 66(3): 443-467
[7] Bourdon P S. Similarity of parts to the whole for certain multiplication operators. Proc Amer Math Soc, 1987, 99(3): 563-567
[8] Chen J, Pau J, Wang M. Essential norms and Schatten(-Herz) classes of integration operators from Bergman spaces to Hardy spaces. Results Math, 2021, 76(2): Art 88
[9] Chen J, Wang M. Integration operators between Bloch type spaces and growth spaces on the unit ball of a complex Banach space. Ann Polon Math, 2021, 127(3): 177-199
[10] Chen J, Wang M. Integration operators on Hardy-Carleson type tent spaces. J Math Anal Appl, 2022, 506(1): Paper No. 125562, 25 pp
[11] Contreras M D, Peláez J Á, Pommerenke C, et al. Integral operators mapping into the space of bounded analytic functions. J Funct Anal, 2016, 271(10): 2899-2943
[12] Cowen C C, MacCluer B D. Composition Operators on Spaces of Analytic Functions. Boca Raton: CRC Press, 1995
[13] Galanopoulos P, Girela D, Peláez J Á. Multipliers and integration operators on Dirichlet spaces. Trans Amer Math Soc, 2011, 363(4): 1855-1886
[14] Girela D, Merchán N. Multipliers and integration operators between conformally invariant spaces. Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM, 2020, 114(4): Art 181
[15] Girela D, Peláez J Á. Carleson measures, multipliers and integration operators for spaces of Dirichlet type. J Funct Anal, 2006, 241(1): 334-358
[16] Kato T. Perturbation theory for nullity, deficiency and other quantities of linear operators. J Analyse Math, 1958, 6: 261-322
[17] Laitila J, Miihkinen S, Nieminen P J. Essential norms and weak compactness of integration operators. Arch Math (Basel), 2011, 97(1): 39-48
[18] Laitila J, Nieminen P J, Saksman E, et al. Compact and weakly compact composition operators on BMOA. Complex Anal Oper Theory, 2013, 7(1): 163-181
[19] Laitila J, Nieminen P J, Saksman E, et al. Rigidity of composition operators on the Hardy space $H^p$. Adv Math, 2017, 319: 610-629
[20] Li P, Liu J, Lou Z.Integral operators on analytic Morrey spaces. Sci China Math, 2014, 57(9): 1961-1974
[21] Lin Q, Liu J, Wu Y. Strict singularity of Volterra type operators on Hardy spaces. J Math Anal Appl, 2020, 492(1): Art 124438
[22] Lin Q, Liu J, Wu Y. Strict singularity of weighted composition operators on derivative Hardy spaces. Oper Matrices, 2021, 15(3): 1171-1180
[23] Lindström M, Miihkinen S, Nieminen P J. Rigidity of weighted composition operators on $H^p$. Ann Acad Sci Fenn Math, 2020, 45(2): 825-828
[24] Lindström M, Palmberg N. Duality of a large family of analytic function spaces. Ann Acad Sci Fenn Math, 2007, 32(1): 251-267
[25] Liu J, Yuan C, Zeng H. Schatten classes of Volterra operators on Bergman-type spaces in the unit ball. Commun Pure Appl Anal, 2022, 21(10): 3425-3439
[26] Luecking D H. Inequalities on Bergman spaces. Illinois J Math, 1981, 25(1): 1-11
[27] MacCluer B, Zhao R. Vanishing logarithmic Carleson measures. Illinois J Math, 2002, 46(2): 507-518
[28] Miihkinen S. Strict singularity of a Volterra-type integral operator on $H^p$. Proc Amer Math Soc, 2017, 145(1): 165-175
[29] Miihkinen S, Nieminen P J, Saksman E, et al. Structural rigidity of generalised Volterra operators on $H^p$. Bull Sci Math, 2018, 148: 1-13
[30] Miihkinen S, Pau J, Perälä A, et al. Volterra type integration operators from Bergman spaces to Hardy spaces. J Funct Anal, 2020, 279(4): Art 108564
[31] Miihkinen S, Pau J, Perälä A, et al.Rigidity of Volterra-type integral operators on Hardy spaces of the unit ball. arXiv: 2004.12671
[32] Pau J. Integration operators between Hardy spaces on the unit ball of $\mathbb{C}^n$. J Funct Anal, 2016, 270(1): 134-176
[33] Pau J, Zhao R. Carleson measures, Riemann-Stieltjes and multiplication operators on a general family of function spaces. Integral Equations Operator Theory, 2014, 78(4): 483-514
[34] Pommerenke C. Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation. Comment Math Helv, 1977, 52(4): 591-602
[35] Siskakis A G, Zhao R. A Volterra type operator on spaces of analytic functions//Jarosz K. Function Spaces: Proceedings of the Third Conference on Function Spaces. Providence: American Mathematical Society, 1999: 299-311
[36] Wang M, Zhou L. Embedding derivatives and integration operators on Hardy type tent spaces. Acta Math Sin Engl Ser, 2022, 38(6): 1069-1093
[37] Xiao J. Riemann-Stieltjes operators on weighted Bloch and Bergman spaces of the unit ball. J London Math Soc, 2004, 70(1): 199-214
[38] Xiao J.The $\mathscr{Q}_p$ Carleson measure problem. Adv Math, 2008, 217(5): 2075-2088
[39] Yang Y, Liu J. Integral operators on Bergman-Morrey spaces. J Geom Anal, 2022, 32(6): Art 181
[40] Zhang X, Li S, Shang Q, et al. An integral estimate and the equivalent norms on $F(p,q,s,k)$ spaces in the unit ball. Acta Math Sci, 2018, 38B(6): 1861-1880
[41] Zhao R. On a general family of function spaces. Ann Acad Sci Fenn Math Diss, 1996, No 105
[42] Zhao R. On logarithmic Carleson measures. Acta Sci Math (Szeged), 2003, 69(3/4): 605-618
[43] Zhao R.On $F(p,q,s)$ spaces. Acta Math Sci, 2021, 41B(6): 1985-2020
[44] Zhu K.Operator Theory in Function Spaces. 2nd ed. Providence: American Mathematical Society, 2007

SOME PROPERTIES OF THE INTEGRATION OPERATORS ON THE SPACES ${F(p,q,s)}$^*

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 10

Metrics

Comments

Recommended 10

[1]	Kehe ZHU. THE BEREZIN TRANSFORM AND ITS APPLICATIONS [J]. Acta mathematica scientia,Series B, 2021, 41(6): 1839-1858.
[2]	Guanlong BAO, Hasi WULAN. $Q_K$ SPACES: A BRIEF AND SELECTIVE SURVEY [J]. Acta mathematica scientia,Series B, 2021, 41(6): 2039-2054.
[3]	Lifang ZHOU, Yun FAN, Jin LU. THE PRECISE NORM OF A CLASS OF FORELLI-RUDIN TYPE OPERATORS ON THE SIEGEL UPPER HALF SPACE [J]. Acta mathematica scientia,Series B, 2021, 41(5): 1537-1546.
[4]	Yali PAN, Qingying XUE. TWO WEIGHT CHARACTERIZATIONS FOR THE MULTILINEAR LOCAL MAXIMAL OPERATORS [J]. Acta mathematica scientia,Series B, 2021, 41(2): 596-608.
[5]	Lifang ZHOU. ON THE BOUNDEDNESS AND THE NORM OF A CLASS OF INTEGRAL OPERATORS [J]. Acta mathematica scientia,Series B, 2015, 35(6): 1475-1482.
[6]	WANG Chuan-Rong. RANDOM SINGULAR INTEGRAL OF RANDOM PROCESS WITH SECOND ORDER MOMENT [J]. Acta mathematica scientia,Series B, 2005, 25(2): 376-384.
[7]	DENG Dong-Gao, YA Li-Xin. COMMUTATORS OF SINGULAR INTEGRAL OPERATORS WITH NON-SMOOTH KERNELS [J]. Acta mathematica scientia,Series B, 2005, 25(1): 137-144.
[8]	GONG Ya-Fang, DU Jin-Yuan. SINGULAR INTEGRAL OPERATORS IN L2 SPACES ASSOCIATED WITH THE GENERALIZED JACOBI WEIGHTS [J]. Acta mathematica scientia,Series B, 2002, 22(4): 484-488.
[9]	GONG Ya-Fang, DU Jin-Yuan. SINGULAR INTEGRAL OPERATORS IN L2 SPACE WITH CHEBYSHEV WEIGHTS [J]. Acta mathematica scientia,Series B, 2000, 20(1): 63-67.
[10]	Yali PAN. SOME CONVERGENCE PROBLEMS REGARDING THE FRACTIONAL SCHRÖDINGER PROPAGATOR ON NONCOMPACT MANIFOLDS^* [J]. Acta mathematica scientia,Series B, 2023, 43(5): 2309-2319.