limit of Riemann-Stieltjes sums as an integral on $\mathscr{H}$

Question

I was reading Leon Takhtajan's book on quantum mechanics and, at some point, he states the J. von Neumann Theorem. The first part of this theorem is as follows.

For every self-adjoint operator $A$ on the Hilbert space $\mathscr{H}$ there exists a unique projection-valued resolution of the identity $P(\lambda)$, satisfying the following properties: (a) $$D(A) = \bigg{\{} \varphi \in \mathscr{H}: \hspace{0.1cm} \int_{-\infty}^{\infty}\lambda^{2}d(P(\lambda)\varphi,\varphi) < +\infty\bigg{\}}$$ and for every $\varphi \in D(A)$ $$A\varphi = \int_{-\infty}^{\infty}\lambda dP(\lambda)\varphi $$ defined as a limit of Riemann-Stieltjes sums in the strong topology on $\mathscr{H}$. The support corresponding projection-valued measure $P$ coincides with the spectrum of the operator $A$: $\lambda \in \sigma(A)$ if, and only if $P_{A}((\lambda-\epsilon,\lambda+\epsilon))\neq 0$ for all $\epsilon > 0$.

What the author mean by "as a limit of Riemann-Stieltjes sums in the strong topology on $\mathscr{H}$"? I don't understand this definition and was not able to find it on the internet. Could someone clarify this definition?

Answer 1

Three comments, but too long:

1. The statement is correct in the case of a bounded, self-adjoint operator. Riemann sums are defined in the usual way, except that they are being used for functions with values in a Hilbert space and this isn´t a significant problem since the Calculus I definition of such sums only uses the vector space structure of the line. The strong topology refers to the topology on the operator space $L(H)$ which is defined by the family of seminorms $T\mapsto \|T\phi\|$ as $\phi$ runs through $H$. For this reason, it is perhaps more transparent to write it is as $$A\phi=\int \lambdaß,dP(\lambda)$$ and work in $L(H)$.

2. In the case of unbounded operators, it is confusing and redundant to refer to the strong topology since there is no such topology on the space of unbounded operators, at least not in the standard literature (but see below). In this case, everything in the formula lives in the Hilbert space which now means that for each $\phi \in D(A)$, the Riemann sums for the inegrals converges in the Hilbert space norm.

3- The third comment is really just icing on the cake, but it might be of interest. In contradiction to what I stated above, there is a strong topology on the space of unbounded (self-adjoint) operators--it just hasn´t found its way into the literature since it involves some non-standard topics (strict topologies). The basic idea is that each (not necessarily bounded) self-adjoint operator induces (via the functional calculus) a mapping $\Phi : f\mapsto f(T)$ from the algebra of bounded, continuous functions on the line to $L(H)$, which is a continuous, linear, multiplicative contraction which preserves the involution and units. The converse is also true--every such mapping is induced in this manner. But here´s the rub- Although the spaces involved are both Banach spaces, the continuity required for the converse to hold is not in terms of these structures but of so-called strict topologies which were introduced in the 50´s of the last century.

This leads to a rather unconventional statement of the spectral theorem, namely that the family of self-adjoint operators on a Hilbert space is naturally identifiable with a generalised spectrum of the algebra of continuous, bounded functions on the line, as described above. This is certainly less direct than the one quoted in your posting, but does have many advantages, particularly with respect to applications in quantum theory and it does provide a way of imposing on the space of observables a respectable topology, a construction which brings many obvious advantages