Opuscula Math. 41, no. 5 (2021), 649-665
https://doi.org/10.7494/OpMath.2021.41.5.649
Opuscula Mathematica
Closed range weighted composition operators between Lp-spaces
Ching-on Lo
Anthony Wai-keung Loh
Abstract. We characterize the closedness of ranges of weighted composition operators between \(L^p\)-spaces, where \(1 \leq p \leq \infty\). When the \(L^p\)-spaces are weighted sequence spaces, several corollaries about this class of operators are also deduced.
Keywords: weighted composition operator, Lebesgue space, closed range.
Mathematics Subject Classification: 47B33, 46E30.
- J.T. Campbell, J.E. Jamison, On some classes of weighted composition operators, Glasgow Math. J. 32 (1990), 87-94.
- J.B. Conway, A Course in Functional Analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990.
- T. Hoover, A. Lambert, J. Quinn, The Markov process determined by a weighted composition operator, Studia Math. 72 (1982), 225-235.
- M.R. Jabbarzadeh, Weighted composition operators between \(L^p\)-spaces, Bull. Korean Math. Soc. 42 (2005), 369-378.
- M.R. Jabbarzadeh, E. Pourreza, A note on weighted composition operators on \(L^p\)-spaces, Bull. Iranian Math. Soc. 29 (2003), 47-54.
- R. Kumar, Weighted composition operators between two \(L^p\)-spaces, Mat. Vesnik 61 (2009), 111-118.
- C.O. Lo, Compact-type weighted composition operators between \(L^p\)-spaces, Banach J. Math. Anal. 14 (2020), 1421-1442.
- C.O. Lo, A.W.K. Loh, Fredholm weighted composition operators, Oper. Matrices 13 (2019), 169-186.
- C.O. Lo, A.W.K. Loh, Compact weighted composition operators between \(L^p\)-spaces, Bull. Aust. Math. Soc. 102 (2020), 151-161.
- R.K. Singh, N.S. Dharmadhikari, Compact and Fredholm composite multiplication operators, Acta Sci. Math. (Szeged) 52 (1988), 437-441.
- R.K. Singh, A. Kumar, Multiplication operators and composition operators with closed ranges, Bull. Aust. Math. Soc. 16 (1977), 247-252.
- H. Takagi, Compact weighted composition operators on \(L^p\), Proc. Amer. Math. Soc. 116 (1992), 505-511.
- H. Takagi, K. Yokouchi, Composition operators between \(L^p\)-spaces, The structure of spaces of analytic and harmonic functions and the theory of operators on them, Kyoto Univ. Res. Inst. Math. Sci. Kokyuroku 946 (1996), 18-24 [in Japanese].
- H. Takagi, K. Yokouchi, Multiplication and composition operators between two v-spaces, Function spaces (Edwardsville, IL, 1998), Contemp. Math., 232, Amer. Math. Soc., Providence, RI, 1999, 321-338.
- Ching-on Lo (corresponding author)
https://orcid.org/0000-0003-2735-8726- Division of Science, Engineering and Health Studies, College of Professional and Continuing Education, The Hong Kong Polytechnic University
- Anthony Wai-keung Loh
- https://orcid.org/0000-0002-2759-3198
- Division of Science, Engineering and Health Studies, College of Professional and Continuing Education, The Hong Kong Polytechnic University
- Communicated by P.A. Cojuhari.
- Received: 2021-07-29.
- Accepted: 2021-08-10.
- Published online: 2021-09-30.
