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.
Categories: math.FA