2024/01/02 by Leonetti, Paolo
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2401.01136
Given an ideal I on ω and a bounded real sequence x, we denote by corex(I) the smallest interval [a,b] such that \n ∈ ω: xn ∉ [a-ε,b+ε]\ ∈ I for all ε>0 (which corresponds to the interval [ \liminf x, \limsup x ] if I is the ideal Fin of finite subsets of ω). First, we characterize all the infinite real matrices A such that coreAx(J)=corex(I) for all bounded sequences x, provided that J is a countably generated ideal on ω and A maps bounded sequences into bounded sequences. Such characterization fails if both I and J are the ideal of asymptotic density zero sets. Next, we show that such equality is possible for distinct ideals I, J, answering an open question in [J.~Math.~Anal.~Appl.~321 (2006), 515--523]. Lastly, we prove that, if J=Fin, the above equality holds for some matrix A if and only if I=Fin or I=Fin⊕ P(ω).