Noncommutative functional calculate and its application

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.