Core equality of real sequences

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 [lim infx,lim supx] 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 is an isomorphic copy of Fin⊕P(ω) on ω.