  Frames and orthonormal bases are naturally linked to bounded operators. To
tackle unbounded operators those sequences might not be well suited. This has
already been noted by von Neumann in the 1920ies. But modern frame theory also
investigates other sequences, including those that are not naturally linked to
bounded operators. The focus of this manuscript will be two such kind of
sequences: lower frame and Riesz-Fischer sequences. We will discuss the
inter-relation of those sequences. We will fill a hole existing in the
literature regarding the classification of those sequences by their synthesis
operator. We will use the idea of generalized frame operator and Gram matrix
and extend it. We will use that to show properties for canonical duals for
lower frame sequences, like e.g. a minimality condition regarding its
coefficients. We will also show that other results that are known for frames
can be generalized to lower frame sequences.
  To be able to tackle these tasks, we had to revisit the concept of
invertibility (in particular for non-closed operators). In addition, we are
able to define a particular adjoint, which is uniquely defined for any
operator.
