# Original statement of Naimark's dilation theorem

Naimark's dilation theorem in papers and textbooks is usually stated as:

Let $E$ be a regular, positive, $B(\mathcal H)$-valued measure on $X$. Then there exists a Hilbert space $\mathcal K$, a bounded linear operator $V: \mathcal H \rightarrow \mathcal K$, and a regular, self-adjoint, spectral $B(\mathcal K)$-valued measure $F$ on $X$, such that $E(B) = V^*F(B)V$ (from Paulsen's book Theorem 4.6).

What was the original formulation of Naimark's dilation theorem? It seems conceivable that it changed over the 70+ years.

Did he assume regularity, or is this assumption coming from the later version of this theorem proved by Stinespring. Were his operator-valued measures weakly countably additive?

My trouble is that I cannot find the original paper:

Does anyone know if this paper is legitimately online anywhere, in Russian or an English translation? As Willie Wong mentions in the comments my fall back will be interlibrary loan.

• Zbmath seems to think it is in English (so does ISI, apparently); also the Doklady from 1935 - 1947 have immediate translations into German French and English published, according to Wikipedia, so you shouldn't be so pessimistic in your final sentence. My university doesn't have a copy, but Stanford does, so Interlibrary loan seems a possibility. Nov 16, 2017 at 20:12
• @WillieWong Thanks for the info. I was hoping for something more immediate than ILL (I think they run it up by dogsled to Winnipeg in the winter) but it's good to know that it probably is a possibility. I really wish that more of these old journals would be digitized. Nov 17, 2017 at 1:27
• Amazingly some of them are. I was poking around on elibrary.ru and a few of the articles from the 1940s in Doklady are actually scanned and available. I am not sure how they decided which ones to scan though. Nov 17, 2017 at 16:18
• The original paper is in Russian: M.A. Naimark, On a Representation of Additive Operator Set Functions (in Russian), Dokl. Akad. Nauk SSSR. 41 (1943), 373--375. Your reference is the English translation. Oct 16, 2019 at 15:00