Geodesic
Applications of ultraproducts in operator theory. A~simple proof of E.~Bishop's theorem
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part IX, Tome 92 (1979), pp. 230-240

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Using technique of ultraproducts of Banach spaces a simple proof of the following E. Bishop's theorem is presented: the closure in the strong operator topology of the set of normal operators on the Hilbert space coincides with the set of subnormal operators. The article contains also some other applications of untraproducts in operator theory (existence of dilation,characterization of spectral measures in Hilbert spaces, similarity of operators, and others). 
@article{ZNSL_1979_92_a13,
     author = {V. V. Peller},
     title = {Applications of ultraproducts in operator theory. {A~simple} proof of {E.~Bishop's} theorem},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {230--240},
     publisher = {mathdoc},
     volume = {92},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1979_92_a13/}
}
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V. V. Peller. Applications of ultraproducts in operator theory. A~simple proof of E.~Bishop's theorem. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part IX, Tome 92 (1979), pp. 230-240. http://geodesic.mathdoc.fr/item/ZNSL_1979_92_a13/