Trespass?

Possession - Section 8 and Section 21

I was sent the below wording by my tenant 6 weeks before the end of the 6 month AST:

"I’m writing this email to let you know that sadly myself and xxx will not be continuing the tenancy in may."

I took this as notice that they would not renew the 6 month tenancy which ended on 29th May and would vacate.

No further rent was paid and Mr x moved straight out. She is now saying this was not notice as no move out date was confirmed. Is this trespass? Would this stand up on court? She had said she was awaiting given temporary secure accommodation which never materialised. I am issuing s.21 anyway but looking if I can do it quicker with trespass? Any advice?
Thank you.