The 53-year-old was jailed for life with a minimum tariff of 20 years in August for stabbing to death Barbara Griffin (59) and seriously injuring Emma Anton (85) in a frenzied night-time attack at Le Geyt flats.
He had denied the offences but was found guilty by a jury following a two-week trial in the Royal Court earlier in the year.
It was the second time that Tregaskis had been tried for the offences, after being acquitted of both charges following a Royal Court trial in 1991.
Tregaskis, who is already serving a separate life sentence for murdering a man in Cornwall in 1997, is seeking leave to appeal against the conviction. He is also appealing against the sentence.
He was due to be represented by Advocate Rebecca Morley-Kirk, with the Crown’s case being led by Solicitor General, Mark Temple.
The case was set to be heard by Sir Wyn Williams, the president of the appeal court panel, former Bailiff Sir William Bailhache and the Bailiff of Guernsey, Richard McMahon.