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 $F_{xx*}(M_{zψ(z)}-a)$ on $\mathcal{L}^{2}(σ(|T+a|),μ_{|T+a|})$ for any given $T\in\mathcal{B}(\mathbb{H})$, where $ψ(z)\in\mathcal{L}^{\infty}(σ(|T+a|),μ_{|T+a|})$, $a\inρ(T)$, $F_{xx*}(f(xx^*))=f(x^*x)$, $\mathcal{B}(\mathbb{H})$ is the set of all bounded linear operator on complex separable Hilbert space $\mathbb{H}$. Also, we get that if $zψ(z)\in Fix(F_{xx^*})$, 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.