There are many references to Church of Kerala established by St.thomas
St.Ephraem, the Syrian (AD 300-373)
St.Ambrose
Eusebius(264-340 )
St.Jerome(345-490)
Marcopolo
Nicolo De Conti
John Chrysostom, ( 347—407 )
Many more reference in liturgy of Churches around the world
You didnt answer my question
There is a translation of a Syriac letter of Mar Gabriel,a Syrian bishop sent to India,please read that.
Chronicles of Malabar: Mission of Mar Simon and Mar Gabriel-An Attempt to reinforce East Syrianism
The earliest available report seems to be of Dionysio. Dionysio ( 1578) gives the arrival of Thomas of Cana after Mar Sabrisho and Mar Piruz ie, after 9th century. Gouvea ( 1604) also dates the arrival in 9th century. The Dutch writer Visscher ( 1743) also gives the date as 745 AD. Hugh gives the date as 780 AD. Assemani in 900 AD. Paolo gives the arrival as 825 AD. Some of the 16th century records doesn’t mention the dates.
Reported byYear Year of arrival
Dionysio 1578 After the 9th Century
Gouvea 1604 9th Century
Visscher 1743 745 AD
Hugh 780 AD
Assemani 900 AD
Paolo 825 AD
Local document 18th Century 345 AD ( also according to current Southist tradition)