Spectral cuts and unconventional functional calculi

In this work, we prove that linear bounded operators $T$ on a Banach space $X$ allowing spectral cuts along rectifiable Jordan curves meeting their spectrum are related to classes of operators admitting an unconventional functional calculus. We identify several such classes and address the consequences regarding the existence of non-trivial closed invariant subspaces, extending previous results of Chalendar. Furthermore, we establish that every operator belonging to a broad subclass of compact perturbations of diagonalizable normal operators on separable Hilbert spaces, namely, trace-class perturbations, possesses an unconventional functional calculus and is super-decomposable, thereby extending earlier results obtained by the authors.