Noncommutative functional calculate and its application

Abstract

In this paper we construct an unitary operator $F_{xx\ast }$ such that $(F_{x{x}^{\ast }}{)}^2=identity$ and $Fix(F_{x{x}^{\ast }})\ne \varnothing$. We get the unitary equivalent representations $F_{xx\ast }(M_{z\psi (z)}-a)$ on ${\mathcal{L}}^2(\sigma (|T+a|),{\mu }_{|T+a|})$ for any given $T\in \mathcal{B}(\mathbb{H})$, where $\psi (z)\in {\mathcal{L}}^{\infty }(\sigma (|T+a|),{\mu }_{|T+a|})$, $a\in \rho (T)$, $F_{xx\ast }(f(x{x}^{\ast }))=f(x^{\ast }x)$, $\mathcal{B}(\mathbb{H})$ is the set of all bounded linear operator on complex separable Hilbert space $\mathbb{H}$. Also, we get that if $z\psi (z)\in Fix(F_{x{x}^{\ast }})$, 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 $T^{\ast -1}$ is.

14pages. arXiv admin note: substantial text overlap with arXiv:1503.06750