Geodesic
Point unitary spectrum of almost unitary operators
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XII, Tome 126 (1983), pp. 143-149

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Let $E$ be a subset of the unit circle $\mathbb T$. There exists an almost unitary operator $L$ such that $E=\sigma_p(L)\cap\mathbb T$ if $E$ is a countable union of Carleson sets. (An operator $L$ is called almost unitary if it is a sum of unitary and nuclear operators). 
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     author = {N. G. Makarov},
     title = {Point unitary spectrum of almost unitary operators},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {143--149},
     publisher = {mathdoc},
     volume = {126},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1983_126_a15/}
}
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N. G. Makarov. Point unitary spectrum of almost unitary operators. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XII, Tome 126 (1983), pp. 143-149. http://geodesic.mathdoc.fr/item/ZNSL_1983_126_a15/