A unifying functional calculus for well-bounded operators: from bounded variation to Borel measurable functions via weak spectral families
DOI:
https://doi.org/10.14419/0cb2s759منشور
07-06-2026الكلمات المفتاحية:
Banach spaces; functional calculus; absolutely continuous functions; bounded variation; well-bounded operators; weak spectral familyالملخص
We develop a single functional calculus that simultaneously encompasses the bounded variation (BV), absolutely continuous (AC), and Borel measurable calculi for well‑bounded operators on Banach spaces. The key ingredient is a weak spectral family $\{E(\lambda)\}\subset B(X^*)$ concentrated on an interval $[a,b]$, which represents the operator $T\in B(X)$ by $\langle Tx, y^*\rangle = b\langle x,y^*\rangle - \int_a^b \langle x, E(\lambda)y^*\rangle \,d\lambda$. For any function $f$ we define $f(T)$ via $\langle f(T)x, y^*\rangle = f(b)\langle x,y^*\rangle - \int_a^b f'(\lambda)\,\langle x, E(\lambda)y^*\rangle\,d\lambda$, whenever the right‑hand side yields a bounded operator. We prove that this definition is always valid for $f\in BV[a,b]$ (the BV calculus). If the weak spectral family is absolutely continuous, the calculus extends to all $f\in AC[a,b]$. If, additionally, the spectral family is countably additive (i.e., comes from a projection‑valued measure) and $X$ is reflexive, the calculus extends to all bounded Borel functions. The paper unifies classical results of Smart, Ringrose and Gillespie, and provides a transparent hierarchy of functional calculi governed solely by the regularity of the weak spectral family.
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