  For H a separable infinite dimensional complex Hilbert space, we prove that
every B(H) operator has a basis with respect to which its matrix representation
has a universal block tridiagonal form with block sizes given by a simple
exponential formula independent of the operator. From this, such a matrix
representation can be further sparsified to slightly sparser forms; it can lead
to a direct sum of even sparser forms reflecting in part some of its reducing
subspace structure; and in the case of operators without invariant subspaces
(if any exists), it gives a plethora of sparser block tridiagonal
representations. An extension to unbounded operators occurs for a certain
domain of definition condition. Moreover this process gives rise to many
different choices of block sizes.
