# lattice of projections

## 1 The Lattice of Projections

In Hilbert spaces there is a bijective   correspondence between closed subspaces and projections (see this entry (http://planetmath.org/ProjectionsAndClosedSubspaces)). This correspondence is given by

 $P\;\longleftrightarrow\;\mathrm{Ran}(P)$

where $P$ is a projection and $\mathrm{Ran}(P)$ denotes the range of $P$.

Since the set of closed subspaces can be partially ordered by inclusion, we can define a partial order $\leq$ in the set of projections using the above correspondence:

 $P\leq Q\;\;\Longleftrightarrow\;\;\mathrm{Ran}(P)\subseteq\mathrm{Ran}(Q)$

But since projections are self-adjoint operators (in fact they are positive operators, as $P=P^{*}P$), they inherit the natural partial ordering of self-adjoint operators (http://planetmath.org/OrderingOfSelfAdjoints), which we denote by $\leq_{sa}$, and whose definition we recall now

 $P\leq_{sa}Q\;\;\Longleftrightarrow\;\;Q-P\text{is a positive operator}$

As the following theorem shows, these two orderings coincide. Thus, we shall not make any more distinctions of notation between them.

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Theorem 1 - Let $P,Q$ be projections in $B(H)$. The following conditions are equivalent      :

• $\mathrm{Ran}(P)\subseteq\mathrm{Ran}(Q)\;\;$ (i.e. $P\leq Q$)

• $QP=P$

• $PQ=P$

• $\|Px\|\leq\|Qx\|$ for all $x\in H$

• $P\leq_{sa}Q$

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Two closed subspaces $Y,Z$ in $H$ have a greatest lower bound  $Y\land Z$ and a least upper bound $Y\lor Z$. Specifically, $Y\land Z$ is precisely the intersection  $Y\cap Z$ and $Y\lor Z$ is precisely the closure    of the subspace    generated by $Y$ and $Z$. Hence, if $P,Q$ are projections in $B(H)$ then $P\land Q$ is the projection onto $\mathrm{Ran}(P)\cap\mathrm{Ran}(Q)$ and $P\lor Q$ is the projection onto the closure of $\mathrm{Ran}(P)+\mathrm{Ran}(Q)$.

The above discussion clarifies that the set of projections in $B(H)$ has a lattice structure. In fact, the set of projections forms a complete lattice, by somewhat as above:

Every family $\{Y_{\alpha}\}$ of closed subspaces in $H$ possesses an infimum  $\bigwedge Y_{\alpha}$ and a supremum  $\bigvee Y_{\alpha}$, which are, respectively, the intersection of all $Y_{\alpha}$ and the closure of the subspace generated by all $Y_{\alpha}$. There is, of course, a correspondent in terms of projections: every family $\{P_{\alpha}\}$ of projections has an infimum $\bigwedge P_{\alpha}$ and a supremum $\bigvee P_{\alpha}$, which are, respectively, the projection onto the intersection of all $\mathrm{Ran}(P_{\alpha})$ and the projection onto the closure of the subspace generated by all $\mathrm{Ran}(P_{\alpha})$.

• We shall see further ahead in this entry, when we discuss orthogonal projections, that the lattice of projections in $B(H)$ is an orthomodular lattice.

## 3 Commuting and Orthogonal Projections

When two projections $P,Q$ commute, the projections $P\land Q$ and $P\lor Q$ can be described algebraically in a very . We shall see at the end of this section   that $P$ and $Q$ commute precisely when its corresponding subspaces $\mathrm{Ran}(P)$ and $\mathrm{Ran}(Q)$ are ”perpendicular    ”.

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Theorem 2 - Let $P,Q$ be commuting projections (i.e. $PQ=QP$), then

• $P\land Q=PQ$

• $P\lor Q=P+Q-PQ$

• $\mathrm{Ran}(P)\lor\mathrm{Ran}(Q)=\mathrm{Ran}(P)+\mathrm{Ran}(Q)$. In particular, $\mathrm{Ran}(P)+\mathrm{Ran}(Q)$ is closed.

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Two projections $P,Q$ are said to be if $P\leq Q^{\perp}$. This is equivalent to say that its corresponding subspaces are orthogonal ($\mathrm{Ran}(P)$ lies in the orthogonal complement of $\mathrm{Ran}(Q)$).

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Corollary 1 - Two projections $P,Q$ are orthogonal if and only if $PQ=0$. When this is so, then $P\lor Q=P+Q$.

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Corollary 2 - Let $P,Q$ be projections in $B(H)$ such that $P\leq Q$. Then $Q-P$ is the projection onto $\mathrm{Ran}(Q)\cap\mathrm{Ran}(P)^{\perp}$.

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We can now see that $P,Q$ commute if and only if $\mathrm{Ran}(P)$ and $\mathrm{Ran}(Q)$ are ”perpendicular”. A somewhat informal and intuitive definition of ”perpendicular” is that of requiring the two subspaces to be orthogonal outside their intersection (this is different of , since orthogonal subspaces do not intersect each other). More rigorously, $P$ and $Q$ commute if and only if the subspaces $\mathrm{Ran}(P)\cap(\mathrm{Ran}(P)\cap\mathrm{Ran}(Q))^{\perp}$ and $\mathrm{Ran}(Q)\cap(\mathrm{Ran}(P)\cap\mathrm{Ran}(Q))^{\perp}$ are orthogonal.

This can be proved using all the above results: The two subspaces are orthogonal iff

 $0=(P-P\land Q)(Q-P\land Q)=PQ-P\land Q$

and $PQ=P\land Q$ iff

 $PQ=P\land Q=(P\land Q)^{*}=(PQ)^{*}=QP$

We can now also see that the lattice of projections is orthomodular: Suppose $P\leq Q$. Then, using the above results,

 $P\lor(Q\land P^{\perp})=P\lor(Q-P)=P+(Q-P)-P(Q-P)=Q$

## 4 Nets of Projections

In the following we discuss some useful and interesting results about convergence and limits of projections.

Let $\Lambda$ be a poset. A net of projections $\{P_{\alpha}\}_{\alpha\in\Lambda}$ is said to be increasing if $\alpha\leq\beta\Longrightarrow P_{\alpha}\leq P_{\beta}$. Decreasing nets are defined similarly.

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Theorem 3 - Let $\{P_{\alpha}\}$ be an increasing net of projections. Then $\lim_{\alpha}P_{\alpha}x=\bigvee_{\alpha}P_{\alpha}\;x$ for every $x\in H$.

In other words, $P_{\alpha}$ converges  to $\bigvee_{\alpha}P_{\alpha}$ in the strong operator topology.

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Similarly for decreasing nets of projections,

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Theorem 4 - Let $\{P_{\alpha}\}$ be a decreasing net of projections. Then $\lim_{\alpha}P_{\alpha}x=\bigwedge_{\alpha}P_{\alpha}\;x$ for every $x\in H$.

In other words, $P_{\alpha}$ converges to $\bigwedge_{\alpha}P_{\alpha}$ in the strong operator topology.

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Theorem 5 - Let $\Lambda$ be a set and $\{P_{\alpha}\}_{\alpha\in\Lambda}$ be a family of pairwise orthogonal projections. Then $\sum P_{\alpha}$ is summable and $\sum P_{\alpha}\,x=\bigvee_{\alpha}P_{\alpha}\;x$ for all $x\in H$.

 Title lattice of projections Canonical name LatticeOfProjections Date of creation 2013-03-22 17:53:29 Last modified on 2013-03-22 17:53:29 Owner asteroid (17536) Last modified by asteroid (17536) Numerical id 14 Author asteroid (17536) Entry type Feature Classification msc 46C07 Classification msc 46C05 Classification msc 06C15 Classification msc 46L10 Synonym projections in Hilbert spaces Related topic OrthomodularLattice Related topic QuantumLogic Related topic ContinuousGeometry Defines minimal projection