nLab projection measure

Idea

Projection (or: projection-valued) measures are operator-valued measures of a special type. They appear for example in the theory of reproducing kernel Hilbert spaces, coherent states and the foundations of quantum mechanics. A projection measure is used to parametrize a complete family of projection operators by subsets of some parameter space.

Given a set $X$ and some $\sigma$-algebra $B$ of subsets of $X$, with $X\in B$, and a complex Hilbert space $H$, a map $P: B\to End H$ is called a projection-(valued) measure on $B$ with values in $End H$ if

• all operators in the image are selfadjoint $P(A) = P(A)^*$

• $P(A_1\cap A_2) = P(A_1) P(A_2)$ for all $A_1,A_2\in B$ where the product is the composition of the operators

• $P(A_1\cup A_2) = P(A_1)+P(A_2)$ for all $A_1,A_2\in B$ such that $A_1\cap A_2 = \emptyset$

• if $A_n\to A$, in the sense of coinciding upper and lower limit of sets, $A= \cap_n \cup_{k\geq n} A_k = \cup_n \cap_{k\geq n} A_k$, then $P(A_n)\to P(A)$ in the strong operator topology. (note: check if strong)

Typical example is that $(X,\tau)$ is a topological space and $B$ is the $\sigma$-algebra $\mathcal{B}(X)$ of Borel subsets of $X$.

References

• Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics. CRC Press (1995) [pdf, gBooks]

• A. A. Kirillov, A. D. Gvišiani, Теоремы и задачи функционального анализа (theorems and exercises in functional analysis), Moskva, Nauka 1979, 1988