Geodesic
Counterexample to a~uniqueness theorem foranalytic operator functions
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XI, Tome 113 (1981), pp. 261-263

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It is proved that there exists a bounded holomorphic operator-function $z\mapsto F(z)$, $|z|1$, with compact values (in a separable Hilbert space) and such that its boundary values $F(\zeta)$, $|\zeta|=1$, are compact on one (given) arc of the circle and not compact on the other. The corresponding example is constructed with the help of vectorial Hankel operators. 
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     author = {D. R. Yafaev},
     title = {Counterexample to a~uniqueness theorem foranalytic operator functions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {261--263},
     publisher = {mathdoc},
     volume = {113},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1981_113_a21/}
}
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D. R. Yafaev. Counterexample to a~uniqueness theorem foranalytic operator functions. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XI, Tome 113 (1981), pp. 261-263. http://geodesic.mathdoc.fr/item/ZNSL_1981_113_a21/