Centered weighted composition operators on L2-spaces are characterized without the product assumption, and spectrally half-centered operators are shown to include unbounded cases whose powers are closed and densely defined.
Giselsson, Half-centered operators, Oper
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Centered weighted composition operators on $L^2$-spaces revisited
Centered weighted composition operators on L2-spaces are characterized without the product assumption, and spectrally half-centered operators are shown to include unbounded cases whose powers are closed and densely defined.