Am I responsible to pay these final bills on my father’s behalf because I was the executor of his trust?

My father passed away early April 2015. I was assigned the executor of my father’s small trust. In late April 2016, I received a letter, addressed to my father, from the hospital requesting payment of a medical bill while my father was in the hospital. Upon receipt of the bill, I contacted the hospital and informed them that my father passed away a year earlier. This was first notice of a bill, and that there was no money in his estate to pay the bill. During the year 2015, I did due diligence in paying all of my father’s bills and waiting until after the anniversary of his death, to ensure no other bills were due, before making final distribution of his small estate. In June 2016, I received another letter, again addressed to my father, from a collection agency requesting payment of the hospital bill. How should this collection process be addressed?