To appear:
The Mathematical Legacy of Victor Lomonosov, De Gruyter, 2020.

We prove every B(H) operator on a separable infinite dimensional complex Hilbert space has a basis for which its matrix is finite block tridiagonal, each with the same fixed precise block sizes given in a simple exponential form. An extension to unbounded operators occurs when a certain domain of definition condition is satisfied. And an extension to finite collections of operators holds, each finite collection with the same block sizes of larger exponential growth depending on the number of operators.