Manuscript Oai280-A@1 · Oai280-A.tex · proposition · 27038823

Proposition: An elementary unbounded-operator separation

Stage 1: three different locality questions › Common-domain commutation versus strong commutation · lines 535–580 · proof included or linked

Oai280-A@1#prop:unbounded

Back to manuscript

There exist self-adjoint operators $P, Q$ which commute on a common dense invariant domain and satisfy a polynomial comparison bound there, but whose spectral projections do not commute.

Proof. On $H = L^{2} (0, 1)$ , set

P = - i \frac{d}{d x}, Dom P = {f \in H^{1} (0, 1) : f (1) = f (0)},

Q = - i \frac{d}{d x}, Dom Q = {f \in H^{1} (0, 1) : f (1) = - f (0)} .

The Fourier bases

e_{n} (x) = e^{2 π in x}, g_{n} (x) = e^{i (2 π n + π) x}

show that $P, Q$ are self-adjoint. On $D = C_{c}^{\infty} (0, 1)$ , both act as the same derivative and preserve $D$ . Hence $P Q f = QP f$ for $f \in D$ .

Let $H_{D} = 1 - \frac{d ^{2}}{d x ^{2}}$ with Dirichlet boundary conditions. For $f \in D$ ,

∥ P f ∥^{2} = ∥ Q f ∥^{2} \leq ⟨ f, H_{D} f ⟩ \leq ∥ f ∥ ∥ H_{D} f ∥ \leq ∥ H_{D} f ∥^{2} .

This is a polynomial comparison bound relative to a positive operator with compact resolvent.

The rank-one spectral projections onto $e_{0}$ and $g_{0}$ do not commute, because

0 < ∣ ⟨ e_{0}, g_{0} ⟩ ∣ = 2/ π < 1.

Rank-one projections commute only when their ranges coincide or are orthogonal. Thus $P, Q$ do not strongly commute.

Displayed formulas in this passage

unlabeled · line 544
$P = - i \frac{d}{d x}, Dom P = {f \in H^{1} (0, 1) : f (1) = f (0)},$
unlabeled · line 548
$Q = - i \frac{d}{d x}, Dom Q = {f \in H^{1} (0, 1) : f (1) = - f (0)} .$
unlabeled · line 553
$e_{n} (x) = e^{2 π in x}, g_{n} (x) = e^{i (2 π n + π) x}$
unlabeled · line 563
$∥ P f ∥^{2} = ∥ Q f ∥^{2} \leq ⟨ f, H_{D} f ⟩ \leq ∥ f ∥ ∥ H_{D} f ∥ \leq ∥ H_{D} f ∥^{2} .$
unlabeled · line 574
$0 < ∣ ⟨ e_{0}, g_{0} ⟩ ∣ = 2/ π < 1.$

Nearby in this section

section: Common-domain commutation versus strong commutation