Applications of ultraproducts in operator theory. A simple proof of E. Bishop's theorem

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).