Definition:Resolution of the Identity

Definition

Let $X$ be a topological space.

Let $\map \BB X$ be the Borel $\sigma$-algebra of $X$.

Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.

Let $\map B {\HH}$ be the space of bounded linear transformations on $\HH$.

Let $\EE : \map \BB X \to \map B {\HH}$ be a function.

For each $x, y \in \HH$, define $\EE_{x, y} : \map \BB X \to \C$ by:

$\map {\EE_{x, y} } A = \innerprod {\map \EE A x} y$

for each $A \in \map \BB X$.

We call $\EE$ a resolution of the identity if and only if:

$(1) \quad$ $\map \EE \O = {\mathbf 0}_{\map B \HH}$ and $\map \EE X = I$

$(2) \quad$ for each $A \in \map \BB X$, $\map \EE A$ is an orthogonal projection on $\HH$

$(3) \quad$ for each $A, B \in \map \BB X$, we have $\map \EE {A \cap B} = \map \EE A \map \EE B$

$(4) \quad$ if $A, B \in \map \BB X$ have $A \cap B = \O$ then $\map \EE {A \cup B} = \map \EE A + \map \EE B$

$(5) \quad$ for each $x, y \in H$, the function $\EE_{x, y}$ is a complex measure.

We call $\EE_{x, y}$ for each $x, y \in \HH$ a scalar-valued measure associated with $\EE$.

Also see

• Results about resolutions of the identity can be found here.

Sources

• 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $12.7$: Definition