# Construct a set of anticommutative operators

Suppose that I have a set of infinitely many operators $\mathbb{A} = \{ A_1, \dots , A_n \}$ with $n \mapsto \infty$. All operators satisfy

$A_i A_j + A_j A_i = 0$ for all $i,j \in \{ 1, \dots, n \}$.

Moreover I assume that $\mathbb{A}$ is a set of linear operators such that these operators can be represented as an (infinite-dimensional) matrix.

Question: Is there a general form of such matrices satisfying above relations? How can I construct these matrices?

• The problem as you state it is not "well-posed". By a general form you probably mean a classification. For that you need additional restrictions. E.g. $A_i$ are self-adjoint bounded operators on a Hilbert space. Otherwise you cannot even classify a couple of such operators. – Yurii Savchuk Jun 18 '15 at 12:57
• I forgot to mention other conditions: These Operators are defined on a Hilbert space and they satisfy $A_i^\dagger=A_i$ for all $i$, i.e. all Operators are hermitean Operators. – kryomaxim Jun 18 '15 at 13:02
• Are they bounded? – Yurii Savchuk Jun 18 '15 at 13:03
• What about an infinite product of Pauli matrices? They anticommute... – draks ... Jun 18 '15 at 13:08
• yes, These Operators are bounded. – kryomaxim Jun 18 '15 at 13:12