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Open AccessArticle

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:HH be a bounded linear operator on a separable Hilbert space H. In this paper, we construct an isomorphism Fxx*:L2(σ(|Ta|),μ|Ta|,ξ)L2(σ(|(Ta)*|),μ|(Ta)*|,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|(Sa)|,ξ(Mzϕ(z)+a)R|Sa|,ξ1 is the noncommutative functional calculus of a normal operator S, where aρ(T), R|Ta|,ξ:L2(σ(|Ta|),μ|Ta|,ξ)H is an isomorphism and Mzϕ(z)+a is a multiplication operator on L2(σ(|Sa|),μ|Sa|,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
Keywords: chaos; invariant subspace; Lebesgue operator; noncommutative functional calculus chaos; invariant subspace; Lebesgue operator; noncommutative functional calculus
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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