arXiv subjects

Lazaro Recht

Publications and source records attributed to Lazaro Recht.

3 recordsLinked to original sources

arXiv2026

Graphs of operators as points in the Grassmann manifold

We study the set $\Gamma$ of graphs of closed, densely defined operators in a Hilbert space $H$, regarded as a subset of the Grassmann manifold $P(H\times H)$ of orthogonal projections in $H\times H$. We show that the subset $\Gamma^b$ of graphs of bounded operators is the open unit ball of $P(H\times H)$ centered at the graph of the zero operator $P_0$ (which projects onto $H\times\{0\}$). This ball is diffeomorphic to $B(H)$ via the map $T\mapsto P_T$ ($=$ the projection onto the graph ${Gr(T)}$ of $T$). We show that graphs of unbounded closed operators lie at the boundary of $\Gamma$. We also study the existence and characteristics of minimal geodesics of $P(H\times H)$ joining two graphs $Gr(A)$, $Gr(B)$. If $A,B$ are selfadjoint, such a geodesic always exists, and we construct explicitly a distinguished exponent using the five-space decomposition associated to the pair of subspaces $Gr(A)$, $Gr(B)$. An explicit low-dimensional example shows that the geodesic joining two graphs need not remain inside $\Gamma$, i.e., does not consist entirely of graphs. We also relate graphs of compact operators to the restricted Grassmannian, and study the problem of common complements for pairs $Gr(S)$, $Gr(T)$, giving positive results when one operator is bounded or under lower boundedness conditions.

math.FA↗

arXiv2011

The left invariant metric in the general linear group

Left invariant metrics induced by the p-norms of the trace in the matrix algebra are studied on the general lineal group. By means of the Euler-Lagrange equations, existence and uniqueness of extremal paths for the length functional are established, and regularity properties of these extremal paths are obtained. Minimizing paths in the group are shown to have a velocity with constant singular values and multiplicity. In several special cases, these geodesic paths are computed explicitly. In particular the Riemannian geodesics, corresponding to the case p=2, are characterized as the product of two one-parameter groups. It is also shown that geodesics are one-parameter groups if and only if the initial velocity is a normal matrix. These results are further extended to the context of compact operators with p-summable spectrum, where a differential equation for the spectral projections of the velocity vector of an extremal path is obtained.

math.DG↗

arXiv2008

Finsler geometry and actions of the p-Schatten unitary groups

Let $p$ be an even positive integer and $U_p(H)$ be the Banach-Lie group of unitary operators $u$ which verify that $u-1$ belongs to the $p$-Schatten ideal $B_p(H)$. Let ${\cal O}$ be a smooth manifold on which $U_p(H)$ acts transitively and smoothly. Then one can endow ${\cal O}$ with a natural Finsler metric in terms of the $p$-Schatten norm and the action of $U_p(H)$. Our main result establishes that for any pair of given initial conditions $$ x\in {\cal O}\hbox{and} X\in (T{\cal O})_x $$ there exists a curve $δ(t)=e^{tz}\cdot x$ in ${\cal O}$, with $z$ a skew-hermitian element in the $p$-Schatten class such that $$ δ(0)=x \hbox{and} \dotδ(0)=X, $$ which remains minimal as long as $t\|z\|_p\le π/4$. Moreover, $δ$ is unique with these properties. We also show that the metric space $({\cal O},d)$ ($d=$ rectifiable distance) is complete. In the process we establish minimality results in the groups $U_p(H)$, and a convexity property for the rectifiable distance. As an example of these spaces, we treat the case of the unitary orbit $$ {\cal O}=\{uAu^*: u\in U_p(H)\} $$ of a self-adjoint operator $A\in B(H)$.

math.DG↗