Given two nets of projections $\{P_i\}$ and $\{Q_i\}$. Both of them tend to $0$. Do we have $P_i\vee Q_i \rightarrow_i 0$?

Question

Given two nets of projections $\{P_i\}$ and $\{Q_i\}$ on a Hilbert space. Both of them decrease to $0$. Do we have $P_i\vee Q_i \rightarrow_i 0$?

Answer 1

EDIT: This approach apparently only works if $P_i$ and $Q_i$ are commuting orthogonal projections.

Note that positivity of operators in a Hilbert space $H$ can be characterised by means of quadratic forms, i.e. $$ T \leq S \iff \forall x \in H \colon \langle T x, x \rangle \leq \langle S x, x \rangle $$

Let $x \in H$ be given and $S_i := P_i \vee Q_i$. Then, $S_i \leq P_i + Q_i$ (THIS FAILS IF $P_i$ and $Q_i$ don't commute!), which gives you $$ \|S_i x\|^2 = \langle S_i x, x \rangle \leq \langle \big(P_i + Q_i\big) x, x \rangle \leq \langle P_i x, x \rangle + \langle Q_i x, x \rangle = \|P_i x\|^2 + \|Q_i x\|^2 \to 0 $$