Noncommutative functional calculate and its application

Abstract: In this paper we construct an unitary operator $F_{xx*}$ such that $(F_{xx^{*}})^2=identity$ and $Fix(F_{xx^*})\neq\emptyset$. We get the unitary equivalent representations $F_{xx*}(M_{z\psi(z)}-a)$ on $\mathcal{L}^{2}(\sigma(|T+a|),\mu_{|T+a|})$ for any given $T\in\mathcal{B}(\mathbb{H})$, where $\psi(z)\in\mathcal{L}^{\infty}(\sigma(|T+a|),\mu_{|T+a|})$, $a\in\rho(T)$, $F_{xx*}(f(xx^*))=f(x^*x)$, $\mathcal{B}(\mathbb{H})$ is the set of all bounded linear operator on complex separable Hilbert space $\mathbb{H}$. Also, we get that if $z\psi(z)\in Fix(F_{xx^*})$, then $T$ has a nontrivial invariant subspace space on $\mathbb{H}$ which has dimension $>1$. Moreover, we define the Lebesgue class $\mathcal{B}_{Leb}(\mathbb{H})\subset\mathcal{B}(\mathbb{H})$ and get that if $T$ is a Lebesgue operator, then $T$ is Li-Yorke chaotic if and only if $T^{*-1}$ is.