A pointwise spectrum and representation of operators

Abstract:

For a self-adjoint operator $A:H\to H$ commuting with an increasing family of projections ${\mathcal {P}}=(P_{t})$ we study the multifunction $t\to \Gamma ^{\mathcal {T}}(t)=\bigcap \{\sigma _{I}:I$ an open set of the topology ${\mathcal {T}}$ containing $t\}$, where $\sigma _{I}$ is the spectrum of $A$ on $P_{I}H$. Let $m_{\mathcal {P}}$ be the measure of maximal spectral type. We study the condition that $\Gamma ^{\mathcal {T}}$ is essentially a singleton, $m_{\mathcal {P}}\{t:\Gamma ^{\mathcal {T}}(t)$ is not a singleton$\}=0$. We show that if ${\mathcal {T}}$ is the density topology and if $m_{\mathcal {P}}$ satisfies the density theorem, in particular if it is absolutely continuous with respect to the Lebesgue measure, then this condition is equivalent to the fact that $A$ is a Borel function of ${\mathcal {P}}$. If ${\mathcal {T}}$ is the usual topology then the condition is equivalent to the fact that $A$ is approched in norm by step functions $\sum \limits _{n\in \mathbb {N}}\Gamma ^{\mathcal {T}} (\alpha _{n})\langle P_{I_{n}} f,f\rangle$, where the set of intervals $\{I_{n}:n\in \mathbb {N}\}$ covers the set where $\Gamma ^{\mathcal {T}}$ is a singleton.