Circled minus sign in Operator theory

Question

I found "circled minus" sign when I read one paper in Operator theory. What does that means? It was $H^{2}\ominus u H^{2}$. What does this mean and where can I find explanation for this?

Answer 1

It's the orthogonal difference. That is, for a subspace $K\subset H$, $$ H\ominus K=\{x\in H:\ x\perp K\}=K^\perp\cap H. $$ When $H$ is the biggest Hilbert space you have around, $H\ominus K$ is simply $K^\perp$; but often times, as in the example you mention, one is considering an inclusion of subspaces, and so the orthogonal difference is the "relative orthogonal complement".