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Esteban Andruchow

Publications and source records attributed to Esteban Andruchow.

At least 19 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↗
arXiv2025

The matched projection and geodesics of the Grassmann manifold

Given an idempotent operator $E$ in a complex Hilbert space ${\mathcal H}$, one can associate to it two orthogonal projections: - The polar decomposition $2E-1=(2P-1)|2E-1|$ provides an orthogonal projection $P$. That the unitary part in the decomposition of $2E-1$ is of this form, i.e., a selfadjoint unitary operator, is a remarkable observation done by G. Corach, H. Porta and L. Recht (see references below). - The question of which, among all orthogonal projections, is the one closest in norm to $E$, provides another projection, the so called {\it matched projection} $m(E)$, which answers this question. It was found by X. Tian, Q. Xu and C. Fu (see references below). In this paper we show that these projections coincide. Moreover, we show that there exists a unique minimal geodesic of the Grassmann manifold of ${\mathcal H}$ (the manifold of closed subspaces of ${\mathcal H}$) that joins $R(E)$ and $R(E^*)$. The orthogonal projection onto the midpoint of this geodesic, also coincides with $m(E)$.

math.FA↗
arXiv2025

The Riemann sphere of a C*-algebra

Given the unital C$^*$-algebra $A$, the unitary orbit of the projector $p_0=\begin{pmatrix}1 & 0 \\ 0 & 0 \end{pmatrix}$ in the C$^*$-algebra $M_2(A)$ of $2\times 2$ matrices with coefficients in $A$ is called in this paper, the Riemann sphere $R$ of $A$. We show that $R$ is a homogeneous reductive C$^\infty$ manifold of the unitary group $U_2(A)\subset M_2(A)$ and carries the differential geometry deduced from this structure (including an invariant Finsler metric). Special attention is paid to the properties of geodesics and the exponential map. If the algebra $A$ is represented in a Hilbert space $H$, in terms of local charts of $R$, elements of the Riemann sphere may be identified with (graphs of) closed operators on $H$ (bounded or unbounded). In the first part of the paper, we develop several geometric aspects of $R$ including a relation between the exponential map of the reductive connection and the cross-ratio of subspaces of $H\times H$. In the last section we show some applications of the geometry of $R$, to the geometry of operators on a Hilbert space. In particular, we define the notion of bounded deformation of an unbounded operator and give some relevant examples.

math.OA↗
arXiv2025

The C$^*$-algebra of a composition reflection

We study the C$^*$ algebra generated by the composition operator $C_a$ acting on the Hardy space $H^2$ of the unit disk, given by $C_af=f\circφ_a$, where $$ φ_a(z)=\frac{a-z}{1-\bar{a}z}, $$ for $|a|<1$. Also several operators related to $C_a$ are examined.

math.OA↗
arXiv2025

Reflections in $L^2(\mathbb{T})$

Let $\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}$ and $\mathbb{T}=\{z\in\mathbb{C}: |z|=1\}$. For $a\in\mathbb{D}$, consider $\varphi_a(z)=\frac{a-z}{1-\bar{a}z}$ and $C_a$ the composition operator in $L^2(\mathbb{T})$ induced by $\varphi_a$: $$ C_a f=f\circ\varphi_a. $$ Clearly $C_a$ satisties $C_a^2=I$, i.e., is a non-selfadjoint reflection. We also consider the following symmetries (selfadjoint reflections) related to $C_a$: $$ R_a=M_{\frac{|k_a|}{\|k_a\|_2}}C_a \ \hbox{ and } \ W_a=M_{\frac{k_a}{\|k_a\|_2}}C_a, $$ where $k_a(z)=\frac{1}{1-\bar{a}z}$ is the Szego kernel. The symmetry $R_a$ is the unitary part in the polar decomposition of $C_a$. We characterize the eigenspaces $N(T_a\pm I)$ for $T_a=C_a, R_a$ or $W_a$, and study their relative positions when one changes the parameter $a$, e.g., $N(T_a\pm I)\cap N(T_b\pm I)$, $N(T_a\pm I)\cap N(T_b\pm I)^\perp$, $N(T_a\pm I)^\perp\cap N(T_b\pm I)$, etc., for $a\ne b\in\mathbb{D}$.

math.FA↗
arXiv2025

Symmetries and reflections from composition operators in the disk

We study the composition operators $C_a$ acting on the Hardy space $H^2$ of the unit disk, given by $C_af=f\circφ_a$, where $$ φ_a(z)=\frac{a-z}{1-\bar{a}z}, $$ for $|a|<1$. These operators are reflections: $C_a^2=1$. We study their eigenspaces $N(C_a\pm 1)$, their relative position (i.e., the intersections between these spaces and their orthogonal complementes for $a\ne b$ in the unit disk) and the symmetries induced by $C_a$ and these eigenspaces.

math.CV↗
arXiv2024

A note on common complements

We discuss the structure of the set $Δ$ consisting of pairs of closed subspaces that have a common complement in a Hilbert space previously studied by Lauzon and Treil (J. Funct. Anal. 212: 500--512, 2004). We prove that $Δ$ is the base space of a real analytic fiber bundle constructed in terms of geometric objects associated to the Grassmann manifold. As a consequence we determine the homotopy type of $Δ$.

math.FA↗
arXiv2024

Subspaces with or without a common complement

Let H be a separable complex Hilbert space. Denote by Gr(H) the Grassmann manifold of H. We study the following sets of pairs of elements in Gr(H): Delta={(S,T) in Gr(H) x Gr(H): there exists Z in Gr(H) such that S\dot{+} Z=T \dot{+} Z=H }, which are pairs of subspaces that have a common complement, and Gamma={(S,T) in Gr(H) x Gr(H): (S,T) does not belong to Delta}, Gamma=Gr(H) x Gr(H) - Delta, which are pairs of subspaces that do not admit a common complement. We identify S withP_S, the subspace S with the orthogonal projection P_S onto S. Thus we may regard Delta and Gamma as subsets of B(H) x B(H) (here B(H) denotes the algebra of bounded linear operators in H. We show that Delta is open, and its connected components are parametrized by the dimension and codimension of the subspaces. The connected component of Delta having both infinite dimensional and co-dimensional subspaces is dense in the corresponding component of Gr(H) x Gr(H). On the other hand, Gamma is a (closed) C^\infty submanifold of B(H) x B(H), and we characterize the connected components of Gamma in terms of dimensions and semi-Fredholm indices. We study the role played by the geodesic structure of the Grassmann geometry of H in the geometry of both Delta and Gamma. Several examples of pairs in Delta and the connected components of Gamma are given in Hilbert spaces of functions.

math.FA↗
arXiv2023

Conjugate points in the Grassmann manifold of a $C^*$-algebra

Let $Gr$ be a component of the Grassmann manifold of a $C^*$-algebra, presented as the unitary orbit of a given orthogonal projection $Gr=Gr(P)$. There are several natural connections in this manifold, and we first show that they all agree (in the presence of a finite trace in $\mathcal A$, when we give $Gr$ the Riemannian metric induced by the Killing form, this is the Levi-Civita connection of the metric). We study the cut locus of $P\in Gr$ for the spectral rectifiable distance, and also the conjugate tangent locus of $P\in Gr$ along a geodesic. Furthermore, for each tangent vector $V$ at $P$, we compute the kernel of the differential of the exponential map of the connection. We exhibit examples where points that are tangent conjugate in the classical setting, fail to be conjugate: in some cases they are not monoconjugate but epinconjugate, and in other cases they are not conjugate at all.

math.FA↗
arXiv2022

The set of partial isometries as a quotient Finsler space

A known general program, designed to endow the quotient space ${\cal U}_{\cal A} / {\cal U}_{\cal B}$ of the unitary groups ${\cal U}_{\cal A}$, ${\cal U}_{\cal B}$ of the C$^*$ algebras ${\cal B}\subset{\cal A}$ with an invariant Finsler metric, is applied to obtain a metric for the space ${\cal I}({\cal H})$ of partial isometries of a Hilbert space ${\cal H}$. ${\cal I}({\cal H})$ is a quotient of the unitary group of ${\cal B}({\cal H})\times{\cal B}({\cal H})$, where ${\cal B}({\cal H})$ is the algebra of bounded linear operators in ${\cal H}$. Under this program, the solution of a linear best approximation problem leads to the computation of minimal geodesics in the quotient space. We find solutions of this best approximation problem, and study properties of the minimal geodesics obtained.

math.FA↗
arXiv2021

Operators which preserve a positive definite inner product

Let ${\cal H}$ be a Hilbert space, $A$ a positive definite operator in ${\cal H}$ and $\langle f,g\rangle_A=\langle Af,g\rangle$, $f,g\in {\cal H}$, the $A$-inner product. This paper studies the geometry of the set $$ {\cal I}_A^a:=\{\hbox{ adjointable isometries for } \langle \ , \ \rangle_A\}. $$ It is proved that ${\cal I}_A^a$ is a submanifold of the Banach algebra of adjointable operators, and a homogeneous space of the group of invertible operators in ${\cal H}$, which are unitaries for the $A$-inner product. Smooth curves in ${\cal I}_A^a$ with given initial conditions, which are minimal for the metric induced by $\langle \ , \ \rangle_A$, are presented. This result depends on an adaptation of M.G. Krein's extension method of symmetric contractions, in order that it works also for symmetrizable transformations (i.e., operators which are selfadjoint for the $A$-inner product).

math.FA↗
arXiv2020

Geodesics of projections in von Neumann algebras

Let ${\cal A}$ be a von Neumann algebra and ${\cal P}_{\cal A}$ the manifold of projections in ${\cal A}$. There is a natural linear connection in ${\cal P}_{\cal A}$, which in the finite dimensional case coincides with the the Levi-Civita connection of the Grassmann manifold of $\mathbb{C}^n$. In this paper we show that two projections $p,q$ can be joined by a geodesic, which has minimal length (with respect to the metric given by the usual norm of ${\cal A}$), if and only if $$ p\wedge q^\perp\sim p^\perp\wedge q, $$ where $\sim$ stands for the Murray-von Neumann equivalence of projections. It is shown that the minimal geodesic is unique if and only if $p\wedge q^\perp= p^\perp\wedge q=0$. If ${\cal A}$ is a finite factor, any pair of projections in the same connected component of ${\cal P}_{\cal A}$ (i.e., with the same trace) can be joined by a minimal geodesic. We explore certain relations with Jones' index theory for subfactors. For instance, it is shown that if ${\cal N}\subset{\cal M}$ are {\bf II}$_1$ factors with finite index $[{\cal M}:{\cal N}]=t^{-1}$, then the geodesic distance $d(e_{\cal N},e_{\cal M})$ between the induced projections $e_{\cal N}$ and $e_{\cal M}$ is $d(e_{\cal N},e_{\cal M})=\arccos(t^{1/2})$.

math.OA↗
arXiv2020

p-Schatten commutators of projections

Let $H=H_+\oplus H_-$ be a fixed orthogonal decomposition of the complex Hilbert space $H$ in two infinite dimensional subspaces. We study the geometry of the set $P^p$ of selfadjoint projections in the Banach algebra $$ {\cal A}^p=\{A\in B(H): [A,E_+]\in B_p(H)\}, $$ where $E_+$ is the projection onto $H_+$ and $B_p(H)$ is the Schatten ideal of $p$-summable operators ($1\le p <\infty$). The norm in ${\cal A}^p$ is defined in terms of the norms of the matrix entries of the operators given by the above decomposition. The space $P^p$ is shown to be a differentiable $C^\infty$ submanifold of ${\cal A}^p$, and a homogeneous space of the group of unitary operators in ${\cal A}^p$. The connected components of $P^p$ are characterized, by means of a partition of $P^p$ in nine classes, four discrete classes and five essential classes: - the first two corresponding to finite rank or co-rank, with the connected components parametrized by theses ranks; - the next two discrete classes carrying a Fredholm index, which parametrizes its components; - the remaining essential classes, which are connected.

math.FA↗
arXiv2020

Grassmann geometry of zero sets in reproducing kernel Hilbert spaces

Let $\mathcal{H}$ be a reproducing kernel Hilbert space of functions on a set $X$. We study the problem of finding a minimal geodesic of the Grassmann manifold of $\mathcal{H}$ that joins two subspaces consisting of functions which vanish on given finite subsets of $X$. We establish a necessary and sufficient condition for existence and uniqueness of geodesics, and we then analyze it in examples. We discuss the relation of the geodesic distance with other known metrics when the mentioned finite subsets are singletons. We find estimates on the upper and lower eigenvalues of the unique self-adjoint operators which define the minimal geodesics, which can be made more precise when the underlying space is the Hardy space. Also for the Hardy space we discuss the existence of geodesics joining subspaces of functions vanishing on infinite subsets of the disk, and we investigate when the product of projections onto this type of subspaces is compact.

math.FA↗
arXiv2020

A note on geodesics of projections in the Calkin algebra

Let ${\cal C}({\cal H})={\cal B}({\cal H}) / {\cal K}({\cal H})$ be the Calkin algebra (${\cal B}({\cal H})$ the algebra of bounded operators on the Hilbert space ${\cal H}$, ${\cal K}({\cal H})$ the ideal of compact operators and $π:{\cal B}({\cal H})\to {\cal C}({\cal H})$ the quotient map), and ${\cal P}_{{\cal C}({\cal H})}$ the differentiable manifold of selfadjoint projections in ${\cal C}({\cal H})$. A projection $p$ in ${\cal C}({\cal H})$ can be lifted to a projection $P\in{\cal B}({\cal H})$: $π(P)=p$. We show that given $p,q \in {\cal P}_{{\cal C}({\cal H})}$, there exists a minimal geodesic of ${\cal P}_{{\cal C}({\cal H})}$ which joins $p$ and $q$ if and only there exist lifting projections $P$ and $Q$ such that either both $N(P-Q\pm 1)$ are finite dimensional, or both infinite dimensional. The minimal geodesic is unique if $p+q- 1$ has trivial anhihilator. Here the assertion that a geodesic is minimal means that it is shorter than any other piecewise smooth curve $γ(t) \in {\cal P}_{{\cal C}({\cal H})}$, $t \in I$, joining the same endpoints, where the length of $γ$ is measured by $\int_I \|\dotγ(t)\| d t$.

math.FA↗
arXiv2019

Unitary operators with decomposable corners

We study pairs $(U,L_0)$, where $U$ is a unitary operator in $H$ and $L_0\subset H$ is a closed subspace, such that $$ P_{L_0}U|_{L_0}:L_0\to L_0 $$ has a singular value decomposition. Abstract characterizations of this condition are given, as well as relations to the geometry of projections and pairs of projections. Several concrete examples are examined.

math.FA↗
arXiv2019

A non commutative Kähler structure on the Poincaré disk of a C*-algebra

We study the Poincaré disk $\d=\{z\in\a: \|z\|<1\}$ of a C$^*$-algebra $\a$ as a homogeneous space under the action of an appropriate Banach-Lie group $\u(θ)$ of $2\times 2$ matrices with entries in $\a$. We define on $\d$ a homogeneous Kähler structure in a non commutative sense. In particular, this Kähler structure defines on $\d$ a homogeneous symplectic structure under the action of $\u(θ)$. This action has a moment map that we explicitly compute. In the presence of a trace in $\a$, we show that the moment map has a convex image when restricted to appropriate subgroups of $\u(θ)$, resembling the classical result of Atiyah-Guillmien-Sternberg.

math.FA↗