Topologically conjugate classification of diagonal operators

2025/05/01 by Xin, Yue, Hou, Bingzhe
#47B99 #54H20 #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Primary 37C15 #Secondary 54C05

paper · doi:10.48550/arxiv.2505.00577

Abstract

Let ℓ^p, 1≤ p<∞, be the Banach space of absolutely p-th power summable sequences and let π_n be the natural projection to the n-th coordinate for n∈ℕ. Let \mathfrakW=\w_n\_n=1^∞ be a bounded sequence of complex numbers. Define the operator D_\mathfrakW: ℓ^p→ℓ^p by, for any x=(x₁,x₂,…)∈ ℓ^p, π_n∘ D_\mathfrakW(x)=w_nx_n for all n≥1. We call D_\mathfrakW a diagonal operator on ℓ^p. In this article, we study the topological conjugate classification of the diagonal operators on ℓ^p. More precisely, we obtained the following results. D_\mathfrakW and D_\vert\mathfrakW\vert are topologically conjugate, where \vert\mathfrakW\vert=\\vert w_n\vert\_n=1^∞. If inf_n\vert w_n\vert>1, then D_\mathfrakW is topologically conjugate to 2I, where I means the identity operator. Similarly, if inf_n\vert w_n\vert>0 and sup_n\vert w_n\vert<1, then D_\mathfrakW is topologically conjugate to (1)/(2)I. In addition, if inf_n\vert w_n\vert=1 and inf_n\vert t_n\vert>1, then D_\mathfrakW and D_\mathfrakT are not topologically conjugate.

Citations

• Topologically conjugate classifications of the translation actions on low-dimensional compact connected Lie groups
• Diagonalization of bosonic quadratic Hamiltonians by Bogoliubov\n transformations
• Orbit equivalence and topological conjugacy of affine actions on compact abelian groups

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