This analysis reveals the characteristics of isometric and quasi-isometric weighted composition operators in Hilbert spaces, indicating implications for spectrum types.
In this paper we characterize m-isometric and quasi-m-isometric weighted composition operators on the Hilbert space L²(μ). Also, we find that normal-m-isometry and normal quasi-m-isometry weighted composition operators have finite spectrum. Consequently we have the results for composition and multiplication operators. In addition, we prove that for m≥ 2, a multiplication operator is m-isometry (quasi-m-isometry) if and only if it is $2$-isometry (quasi-$2$-isometry). Some examples are provided to illustrate our results.
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Ghafri et al. (2025) studied this question.
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