You won't be able to prove the converse because it is not true.

On the Hilbert space , the unilateral shift is the operator T that takes the sequence to . Its adjoint is the backwards shift operator S that takes to (in other words, it shifts the sequence backwards, with the first coordinate falling off the end). Neither T nor S is invertible (see below), but the product ST is the identity operator, which obviously is invertible.

Notice that if you take the product in the reverse order, TS is not the identity. In fact TS takes to . If you shift forwards and then backwards then you end up where you started from, but if you shift backwards and then forwards you lose the first coordinate and cannot retrieve it.

To see why S and T are not invertible, let . Then , so S is not invertible. Also, is not in the range of T, so T is not invertible.