Spectrally $n$-centered operators. I

We introduce the concept of spectrally $n$-centered operators and study it within the framework of unbounded weighted composition operators in $L^2$-spaces. Based on this, we establish a characterisation of spectrally centered unbounded weighted composition operators, generalising Giselsson’s criterion to the unbounded regime. Finally, we analyse various associated conditions, highlighting the delicate measure-theoretic anomalies.