Multiplication operators between mixed norm Lebesgue spaces

Year 2019, Volume: 48 Issue: 3, 771 - 778, 15.06.2019

Héctor Camilo Chaparro

Abstract

The boundedness, compactness and closed range of the multiplication operator defined on mixed norm Lebesgue spaces are characterized in this paper.

Keywords

compact operator , multiplication operator , mixed norm space

References

• Y.A. Abramovich and C.D. Aliprantis, An invitation to operator theory, Graduate Studies in Mathematics, 50, American Mathematical Society, Providence, RI, 2002.
• S.C. Arora, G. Datt, and S. Verma, Multiplication operators on Lorentz spaces, Indian J. Math. 48 (3), 317-329, 2006.
• F.R. Astudillo-Villalba and J.C. Ramos-Fernández, Multiplication operators on the space of functions of bounded variation, Demonstr. Math. 50 (1), 105-115, 2017.
• A. Benedek and R. Panzone, The spaces $L^P$, with mixed norm, Duke Math. J. 28, 301-324, 1961.
• A.P. Blozinski, Multivariate rearrangements and Banach function spaces with mixed norms, Trans. Amer. Math. Soc. 263 (1), 149-167, 1981.
• R.E. Castillo, H.C. Chaparro, and J.C. Ramos Fernández, Orlicz-Lorentz spaces and their multiplication operators, Hacet. J. Math. Stat. 44 (5), 991-1009, 2015.
• R.E. Castillo, J.C. Ramos Fernández, and H. Rafeiro, Multiplication operators in variable Lebesgue spaces, Rev. Colombiana Mat. 49 (2), 293-305, 2015.
• R.E. Castillo, J.C. Ramos Fernández and M. Salas-Brown, The essential norm of multiplication operators on Lorentz sequence spaces, Real Anal. Exchange 41 (1), 245-251, 2016.
• R.E. Castillo, F.A. Vallejo Narváez and J.C. Ramos Fernández, Multiplication and composition operators on weak $L_p$ spaces, Bull. Malays. Math. Sci. Soc. 38 (3), 927- 973, 2015.
• H.C. Chaparro, On Multipliers between Bounded Variation Spaces, Ann. Funct. Anal. 9 (3), 376-383, 2018.
• W. Grey and G. Sinnamon, The inclusion problem for mixed norm spaces, Trans. Amer. Math. Soc. 368 (12), 8715-8736, 2016.
• P.R. Halmos, What does the spectral theorem say?, Amer. Math. Monthly 70, 241-247, 1963.
• H. Hudzik, R. Kumar and R. Kumar, Matrix multiplication operators on Banach function spaces, Proc. Indian Acad. Sci. Math. Sci. 116 (1), 71-81, 2006.
• V. Kolyada and J. Soria, Mixed norms and iterated rearrangements, Z. Anal. Anwend. 35 (2), 119-138, 2016.
• M. Mursaleen, A. Aghajani and K. Raj, Multiplication operators on Cesàro function spaces, Filomat 30 (5), 1175-1184, 2016.
• J.C. Ramos-Fernández, Some properties of multiplication operators acting on Banach spaces of measurable functions, Bol. Mat. 23 (2), 221-237, 2016.
• J.C. Ramos-Fernández and M. Salas-Brown, On multiplication operators acting on Köthe sequence spaces Afr. Mat. 28 (3-4), 661-667, 2017.
• H. Takagi and K. Yokouchi, Multiplication and composition operators between two $L^p$-spaces, Function spaces (Edwardsville, IL, 1998), Contemp. Math. 232, Amer. Math. Soc., Providence, RI, 1999, 321-338.