was it ever colonised by the scandinavians?
Yes, during the late 8th and 9th centuries.
Shetland
 Shetland was colonised during the late 8th and 9th centuries, the fate of the existing indigenous population being uncertain, though it is now seen that Modern Shetlanders have overwhelmingly Scottish DNA according to a major research project, which shows that that only 20 per cent of isles DNA is traceable to Norwegian ancestors. 

what other kingdom claimed ownership of the northern islands of scotland Kingdom of Scotland and Kingdom of Great Britain. can you name the islands that form the above mentioned group? Orkney and Shetland. who were the earliest settlers here? UNANSWERABLE what is the economy of the latter dependent on? Agriculture, aquaculture, fishing and renewable energy.
wiki:19683680
Shetland [SEP] Scandinavian colonisation
In response, Norwegian king Harald Hårfagre ("Harald Fair Hair") annexed the Northern Isles (comprising Orkney and Shetland) in 875. Rognvald Eysteinsson received Orkney and Shetland from Harald as an earldom as reparation for the death of his son in battle in Scotland, and then passed the earldom on to his brother Sigurd the Mighty. The islands converted to Christianity in the late 10th century. King Olav Tryggvasson summoned the "jarl" Sigurd the Stout during a visit to Orkney and said, "I order you and all your subjects to be baptised. If you refuse, I'll have you killed on the spot and I swear I will ravage every island with fire and steel." Unsurprisingly, Sigurd agreed and the islands became Christian at a stroke.