Gerald Teschl in his book "Mathematical Methods in Quantum Mechanics" defines a reducing subspace of an unbounded operator the following way:
Let $T$ be an unbounded operator on a complex Hilbert $H$ space with domain $D(T)\subseteq H$. A closed subspace $A\subseteq H$ is said to be a reducing subspace of $T$ if $P_AT\subseteq TP_A$, where $P_A$ is the orthogonal projection onto $A$.
I can't really get behind this definition. What does this intuitively mean? Is there a relation to invariant subspaces?