Compression Covariance and Tangent kernels

Abstract

Let $A\ge 0$ 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 $C_t=P{T}_{t}P{|}_{PH}$, $V_t=P^{\perp }{T}_{t}P{|}_{PH}$, 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 $E_{s,t}:=V_s^{\ast }{V}_t$. 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.