  The principal theorem of Sz.-Nagy on dilation of a positive definite Hilbert
space operator valued function has played a central role in the development of
the non-self-adjoint operator theory. In this paper we introduce the positive
definiteness for locally Hilbert space operator valued kernels, we prove an
analogue of the Sz.-Nagy dilation theorem and, as application, we obtain
dilation results for locally contractions and locally $\rho$ - contractions as
well as for locally semi-spectral measures.

Functional Analysis