On the approximation of identity in additive-invariant Banach function spaces

Analog of Korovkin type theorem for the sequence of convolution operators in the subspaces of additive-invariant Banach spaces generated by the shift operator is obtained in this work. Convergence to the identity of the sequence of convolution operators with a positive approximate identity is proved. We also prove the convergence to the identity of Steklov operators in Marcinkiewicz spaces.