Minimal curves in U(n) and P(n) with respect to the spectral and the trace norms.

STOJANOFF, DEMETRIO; GHIGLIONI, EDUARDO MARIO; ANTEZANA, JORGE ABEL

Suwon

Workshop; Workshop on barycenters, convexity on metric spaces and positive operators; 2018

Sungkyunkwan University

Consider the Lie group of $n\times n$ complex unitary matrices $\matu$ endowed with the bi-invariant Finsler metric given by the spectral norm,$$\sub{\|X\|}{U}=\|U^*X\|_{\infty}=\|X\|_{\infty}$$for any $X$ tangent to a unitary operator $U$. Given two points in $\matu$, in general there exists infinitely many curves of minimal length. In this talk we will provide a complete description of such curves. As a consequence of this description, we conclude that there is a unique curve of minimal length between $U$ and $V$ if and only if the spectrum of $U^*V$ is contained in a set of the form $\{e^{i\theta},e^{-i\theta}\}$ for some $\theta\in [0,\pi)$. This is similar to the result obtained by Lim in \cite{L1} in the case ofpositive operators endowed with the Thompson metric. Let's recall that the Grassmannian can be modeled as a submanifold of the unitary group (identifying a subspace with the associated orthogonal symmetry). Using this idea we also describe all minimal curves connecting two projections $P$ and $Q$ such that $\left\|P - Q\right\|_{\infty} < 1$. \newline \newline Now consider the cone of $n \times n$ positive invertible matrices $\matppos$endowed with the bi-invariant Finsler metric given by the trace norm,$$\sub{\|X\|}{1,A}=\|A^{-1/2}XA^{-1/2}\|_1$$for any $X$ tangent to $A\in\matppos$. In this context, given two points $A, B \in \matppos$, there also exists infinitely many curves of minimal length. In this case we get the characterization by lifting the problem to the space of hermitian matrices $\matsa$. So, firstly we provide a characterization of the minimal paths joining $X,Y\in \matsa$, if the length of a curve $\alpha:[a,b]\to\matsa$ is measure by$$L(\alpha)=\int_a^b \|\dot{\alpha}(t)\|_1\,dt.$$Once the characterization of the minimal curves is given for $\matsa$, the characterization in $\matppos$ can be obtained using the Exponential Metric Increasing property. On the other hand for $\matu$ the above lifting argument also works in one direction. Indeed, using the same idea as in the case of $\matppos$, we prove that a minimal curve in $\matsa$ leads to a minimal curve in $\matu$ by means of the exponential map. So our characterization in $\mathcal{H}(n)$ give us a way to construct minimal paths in the group of unitary matrices $\matu$.\newline \newline In \cite{L1} Lim also studied the sets of midpoints$$\eme_t(A,B)=\big\{C\in\matppos: d_\infty(A,C)=t\,d_\infty(A,B),\ d_\infty(C,B)= (1-t)\,d_\infty(A,B)\big\}.$$In this talk we will study the set of midpoints in all the previous contexts. We will prove that this set is geodesically convex for $\matppos$ for any unitarily invariant norm. This was already prove in \cite{L1} but we will use a different technique. The same idea turns out to work to prove that the set of midpoints set is geodesically convex for $\mathcal{H}(n)$ endowed with any unitarily invariant norm and for the spectral norm between two unitary matrices $U$ and $V$ provided $\|U-V\|_{\infty}