Operators on Hilbert spaces and essentially bounded functions.

Question

Are there some relationships between operators on Hilbert spaces (linear, bounded, self-adjoint, positive) and essentially bounded functions? Suggestions of books and websites, as well as any explanations, are welcome.

Answer 1

One direction. Let our hilbert space be $H = L^2(X,\mathcal F, \mu)$. Let $\phi : X \to \mathbb C$ be a measurable function. Define the "multiplication" operator $T$ on $H$ by $$ T(f)(x) = \phi(x)\;f(x) $$ Then: $T$ is a bounded operator iff $\phi$ is an essentially bounded function. Operator $T$ is self-adjoint iff almost all values of $\phi$ are real. Operator $T$ is positive iff for almost all $x$, $\phi(x) \ge 0$.