Definition 4

Let $u$ be a normal partial operator on a complex Hilbert space E. Let $x$ and $y$ be in E. The bounded measure on Sp($u$) defined by $f\mapsto \langle x|f(u)y\rangle$ for $f\in \mathscr{K}$ (Sp($u$)) is called the spectral measure of $(x, y)$ relative to $u$. When $x=y$, one says that it is the spectral measure of $x$ relative to $u$.