The Fourier transform converts convolution to pointwise multiplication. By Parseval's theorem, it also transforms the sequence space to the space of square-integrable functions on the unit circle. Under this isometric isomorphism, the convolution operator A gets mapped to the operator of multiplication by f. The adjoint of is the multiplication operator corresponding to the complex conjugate of f. Since multiplication operators commute with each other, it is clear that commutes with its adjoint, hence so does A (so A is normal).
The spectrum of a multiplication operator is the range of the function, so .
The norm of a multiplication operator is the sup norm of the function, so .