抄録
In this work we provide a counterexample for Schur’s Theorem on triangular
matrices on infinite dimensional spaces. Moreover, the counterexample provided is a
compact quasinilpotent operator. Indeed, the result neither depends on the index of
the chosen basis for the matrix representations nor on the upper-lower choice for the
triangular matrix. As a consequence, we see the optimality of a result by Halmos on
matrix representations of operators. Namely, Halmos proved that each operator can
be represented by a matrix with finite columns. Finally, we ‘answer’ a philosophical
question posed by J. B. Conway in [1, p.213].
