What I understand is the following:
FdS-Ferrovie della Sardegna (Railways of Sardinia), founded in 1989, is a narrow gauge railway system using diesel locomotives and multiple units, electrified tram-trains in the metropolitan areas of Sassari and Cagliari and vintage diesel and steam locomotives on the tourist lines. So far till 2010 it was an Italian public company that managed the regional railway network in the island of Sardinia, then in 2010 it was totally integrated into the main regional public transport company known as ARST-Azienda Regionale Sarda Trasporti.
Under the name of ARST is still today the second railway operator in Sardinia, after Ferrovie dello Stato, having 203 kilometers of railways used for public transport, plus another four tourist lines, known as Trenino Verde, which run through the wildest parts of the island.
However the line Cagliari-Olbia (the dark green line in the map below) is operated by Trenitalia S.p.A. that belongs 100% to Ferrovie dello Stato S.p.A.
Also, as you can see, the blue line between Cagliari and Isili is a reduced gauge railway and as such it's very likely NOT to be part of Trenitalia.