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COMPACT WEIGHTED COMPOSITION OPERATORS BETWEEN $L^{p}$-SPACES

Part of: Special classes of linear operators Classical measure theory

Published online by Cambridge University Press:  16 January 2020

CHING-ON LO Open the ORCID record for CHING-ON LO [Opens in a new window]  and
ANTHONY WAI-KEUNG LOH Open the ORCID record for ANTHONY WAI-KEUNG LOH [Opens in a new window]
CHING-ON LO*
Affiliation:
Division of Science, Engineering and Health Studies, College of Professional and Continuing Education, The Hong Kong Polytechnic University, Hong Kong email [email protected]
ANTHONY WAI-KEUNG LOH
Affiliation:
Division of Science, Engineering and Health Studies, College of Professional and Continuing Education, The Hong Kong Polytechnic University, Hong Kong email [email protected]
*
✉[email protected]
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Abstract

We provide complete characterisations for the compactness of weighted composition operators between two distinct $L^{p}$-spaces, where $1\leq p\leq \infty$. As a corollary, when the underlying measure space is nonatomic, the only compact weighted composition map between $L^{p}$-spaces is the zero operator.

Keywords

weighted composition operatorsLebesgue spacescompact operators

MSC classification

Primary: 47B33: Composition operators
Secondary: 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 1 , August 2020 , pp. 151 - 161
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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References

Campbell, J. T. and Jamison, J. E., ‘On some classes of weighted composition operators’, Glasg. Math. J. 32 (1990), 87–94.10.1017/S0017089500009095CrossRefGoogle Scholar
Chan, J. T., ‘A note on compact weighted composition operators on L p(𝜇)’, Acta Sci. Math. (Szeged) 56 (1992), 165–168.Google Scholar
Jabbarzadeh, M. R., ‘Weighted composition operators between L p-spaces’, Bull. Korean Math. Soc. 42 (2005), 369–378.CrossRefGoogle Scholar
Jabbarzadeh, M. R. and Pourreza, E., ‘A note on weighted composition operators on L p-spaces’, Bull. Iranian Math. Soc. 29 (2003), 47–54.Google Scholar
Kumar, R., ‘Weighted composition operators between two L p-spaces’, Mat. Vesnik 61 (2009), 111–118.Google Scholar
Lo, C. O. and Loh, A. W. K., ‘Compact weighted composition operators between L ∞-spaces’, Int. J. Contemp. Math. Sci. 13 (2018), 169–176.10.12988/ijcms.2018.8618CrossRefGoogle Scholar
Narita, K. and Takagi, H., ‘Compact composition operators between L p-spaces. Harmonic/analytic function spaces and linear operators’, Kyoto Univ. Res. Inst. Math. Sci. Kokyuroku 1049 (1998), 129–136 (in Japanese).Google Scholar
Singh, R. K. and Dharmadhikari, N. S., ‘Compact and Fredholm composite multiplication operators’, Acta Sci. Math. (Szeged) 52 (1988), 437–441.Google Scholar
Takagi, H., ‘Compact weighted composition operators on L p’, Proc. Amer. Math. Soc. 116 (1992), 505–511.Google Scholar