On the size of certain subsets of invariant Banach sequence spaces

The essence of the notion of lineability and spaceability is to find linear structures in somewhat chaotic environments. The existing methods, in general, use $\mathit{\text{ad}}\mathit{\text{hoc}}$ arguments and few general techniques are known. Motivated by the search of general methods, in this paper we formally extend recent results of G.\ Botelho and V.V. F'{a}varo on invariant sequence spaces to a more general setting. Our main results show that some subsets of invariant sequence spaces contain, up to the null vector, a closed infinite-dimensional subspace.