Compression Covariance and Tangent kernels

Abstract

Let $A\geq0$ be self-adjoint on a Hilbert space $H$, let $T_{t}=e^{-tA}$, and let $P$ be an orthogonal projection. Relative to the decomposition $H=PH\oplus P^{\perp}H$, write \[ T_{t}=\begin{pmatrix}C_{t} & V^{*}_{t}\\ V_{t} & D_{t} \end{pmatrix}, \] where $C_{t}=PT_{t}P|_{PH}$, $V_{t}=P^{\perp}T_{t}P|_{PH}$, and $D_{t}=P^{\perp}T_{t}P^{\perp}|_{P^{\perp}H}$. The compressed family $\left(C_{t}\right)$ consists of positive contractions but need not form a semigroup. Its defect is given by \[ C_{s+t}-C_{s}C_{t}=V^{*}_{s}V_{t} \] while the complementary block satisfies \[ D_{s+t}-D_{s}D_{t}=V_{s}V^{*}_{t}. \] Thus the failure of $\left\{ C_{t}\right\} $ and $\left\{ D_{t}\right\} $ to be semigroups gives two Gram kernels associated with the same off-diagonal maps. We treat these covariance defects as positive definite operator-valued kernels and use their Kolmogorov spaces to recover the hidden dynamics they encode. We then study short-time rescalings of $E_{s,t}:=V^{*}_{s}V_{t}$. The tangent kernel \[ F\left(s,t\right):=\lim_{\varepsilon\downarrow0}a\left(\varepsilon\right)^{-1}E_{\varepsilon s,\varepsilon t} \] has its own Kolmogorov space, and the lower-right block dynamics induces a positive self-adjoint contraction semigroup on it. The representing vectors of $F$ then satisfy an additive cocycle identity for this semigroup. This gives an intrinsic restriction on the positive kernels that can arise as short-time compression covariance tangents.