Metric geodesics of isometries in a Hilbert space and the extension problem

Abstract:

We consider the problem of finding short smooth curves of isometries in a Hilbert space $\mathcal {H}$. The length of a smooth curve $\gamma (t)$, $t\in [0,1]$, is measured by means of $\int _0^1 \|\dot {\gamma }(t)\|\ dt$, where $\|\ \|$ denotes the usual norm of operators. The initial value problem is solved: for any isometry $V_0$ and each tangent vector at $V_0$ (which is an operator of the form $iXV_0$ with $X^*=X$) with norm less than or equal to $\pi$, there exist curves of the form $e^{itZ}V_0$, with initial velocity $iZV_0=iXV_0$, which are short along their path. These curves, which we call metric geodesics, need not be unique, and correspond to the so called extension problem considered by M.G. Krein and others: in our context, given a symmetric operator \[ X_0|_{R(V_0)}:R(V_0)\to \mathcal {H}, \] find all possible $Z^*=Z$ extending $X_0|_{R(V_0)}$ to all $\mathcal {H}$, with $\|Z\|=\|X_0\|$. We also consider the problem of finding metric geodesics joining two given isometries $V_0$ and $V_1$. It is well known that if there exists a continuous path joining $V_0$ and $V_1$, then both ranges have the same codimension. We show that if this number is finite, then there exist metric geodesics joining $V_0$ and $V_1$.