Generalization of the Riesz-Markov-Kakutani Representation Theorem and Weak Spectral Families

Abstract

Firstly, we generalize some definitions such as the definitions of the weak spectral family, the solitary operator, and the construction of the functional calculus. Secondly, we prove that for a functional calculus $\Phi $ on the measurable space $\left(Z,\; \Sigma \right)$ exists a measurable space $\left(\Omega ,\mathrm{F},\mu \right)$, an operator $U\; :\; X\to L^{p} \left(\Omega ,\mathrm{F},\mu \right)$, and a continuous $\mathrm{\ast }$ -homomorphism $F\; :\; M\left(Z,\; \Sigma \right)\to M\left(\Omega ,\mathrm{F}\right)$, such that $M_{F\left(f\right)} =U^{-1} \Phi \left(f\right)U$ for all $f\in M\left(Z,\, \Sigma \right)$. Thirdly, we establish the correlation between the well-bounded operators and the weak spectral families. It has been proven that for a linear well-bounded operator $A\in L\left(X\right)$ there is a weak spectral family $\left\{E\left(\lambda \right)\in L\left(X^{*} \right),\quad \lambda \in {\mathbb R}\right\}$ on a compact interval $\left[a,\; b\right]$ such that an integral representation $\left\langle A\left(x\right),\; y^{*} \right\rangle =b\left\langle x,\; y^{*} \right\rangle -\int _{\left[a,\; b\right]}\left\langle x,\; E\left(\lambda \right)y^{*} \right\rangle d\lambda $ holds for all $x\in X$, $y^{*} \in X^{*} $, where equivalence is understood in the weak topology.