We treat the Toda lattice equation with operator methods and derive an explicit solution formula in terms of determinants. As an application, we investigate solutions which are given by special settings. In the finite-dimensional case matrices in Jordan canonical form give rise to a new class of solutions. Within this class the well-known N-soliton solutions can be recovered by the special choice of diagonal matrices. Moreover, using diagonal operators we get solutions depending on an infinite number of parameters. We comprehensively discuss the case involving diagonal operators and show that it can be reduced to a very particular situation.