Geodesic
The problem of spectral synthesis for operators with point spectrum
Izvestiya. Mathematics , Tome 4 (1970) no. 3, pp. 670-696

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This article considers the problem of spectral synthesis for operators with a complete system of root vectors. Criteria for the existence of spectral synthesis are given. A criterion for spectral synthesis in terms of geometrical properties of the system of eigenvectors is established for operators with simple spectrum. The relationship between the problems of spectral synthesis for operators $A$ and $A^*$ is studied. 
@article{IM2_1970_4_3_a10,
     author = {A. S. Markus},
     title = {The problem of spectral synthesis for operators with point spectrum},
     journal = {Izvestiya. Mathematics },
     pages = {670--696},
     publisher = {mathdoc},
     volume = {4},
     number = {3},
     year = {1970},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1970_4_3_a10/}
}
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A. S. Markus. The problem of spectral synthesis for operators with point spectrum. Izvestiya. Mathematics , Tome 4 (1970) no. 3, pp. 670-696. http://geodesic.mathdoc.fr/item/IM2_1970_4_3_a10/