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# Noncommutative Functional Calculus and Its Applications on Invariant Subspace and Chaos

by 1,2,3
1
School of Mathematical Sciences, Fudan University, Shanghai 200433, China
2
School of Mathematics, Jilin University, Changchun 130012, China
3
School of Mathematics and Statistics, Xidian University, Xi’an 710071, China
Mathematics 2020, 8(9), 1544; https://doi.org/10.3390/math8091544
Received: 5 August 2020 / Revised: 5 September 2020 / Accepted: 7 September 2020 / Published: 9 September 2020
(This article belongs to the Special Issue Functional Analysis, Topology and Quantum Mechanics)
Let $T:H→H$ be a bounded linear operator on a separable Hilbert space $H$. In this paper, we construct an isomorphism $Fxx*:L2(σ(|T−a|),μ|T−a|,ξ)→L2(σ(|(T−a)*|),μ|(T−a)*|,Fxx*Hξ)$ such that $(Fxx*)2=identity$ and $Fxx*H$ is a unitary operator on $H$ associated with $Fxx*$. With this construction, we obtain a noncommutative functional calculus for the operator T and $Fxx*=identity$ is the special case for normal operators, such that $S=R|(S−a)|,ξ(Mzϕ(z)+a)R|S−a|,ξ−1$ is the noncommutative functional calculus of a normal operator S, where $a∈ρ(T)$, $R|T−a|,ξ:L2(σ(|T−a|),μ|T−a|,ξ)→H$ is an isomorphism and $Mzϕ(z)+a$ is a multiplication operator on $L2(σ(|S−a|),μ|S−a|,ξ)$. Moreover, by $Fxx*$ we give a sufficient condition to the invariant subspace problem and we present the Lebesgue class $BLeb(H)⊂B(H)$ such that T is Li-Yorke chaotic if and only if $T*−1$ is for a Lebesgue operator T. View Full-Text
MDPI and ACS Style

Luo, L. Noncommutative Functional Calculus and Its Applications on Invariant Subspace and Chaos. Mathematics 2020, 8, 1544. https://doi.org/10.3390/math8091544

AMA Style

Luo L. Noncommutative Functional Calculus and Its Applications on Invariant Subspace and Chaos. Mathematics. 2020; 8(9):1544. https://doi.org/10.3390/math8091544

Chicago/Turabian Style

Luo, Lvlin. 2020. "Noncommutative Functional Calculus and Its Applications on Invariant Subspace and Chaos" Mathematics 8, no. 9: 1544. https://doi.org/10.3390/math8091544

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