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Gustavo Corach

Publications and source records attributed to Gustavo Corach.

At least 19 recordsLinked to original sources
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↗
arXiv2023

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 $C_af=f\circ\varphi_a$, where $$ \varphi_a(z)=\frac{a-z}{1-\bar{a}z}, $$ 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↗
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↗
arXiv2018

Projective geometry in the Poincaré disk of a $C^*$-algebra

We study the Poincaré disk ${\cal D}=\{a\in {\cal A}: \|a\|<1\}$ of a C$^*$-algebra ${\cal A}$ from a projective point of view: ${\cal D}$ is regarded as an open subset of the projective line $\mathbb{P}_1{\cal A}$, the space of complemented rank one submodules of ${\cal A}^2$. We introduce the concept of cross ratio of four points in $\mathbb{P}_1{\cal A}$. Our main result establishes the relation between the exponential map $Exp_{z_0}(z_1)$ of ${\cal D}$ ($z_0,z_1\in {\cal D}$) and the cross ratio of the four-tuple $$ δ(-\infty), δ(0)=z_0, δ(1)=z_1 , δ(+\infty), $$ where $δ$ is the unique geodesic of ${\cal D}$ joining $z_0$ and $z_1$ at times $t=0$ and $t=1$, respectively.

math.OA↗
arXiv2018

Projections with fixed difference: a Hopf-Rinow theorem

The set $D_{A_0}$, of pairs of orthogonal projections $(P,Q)$ in generic position with fixed difference $P-Q=A_0$, is shown to be a homogeneus smooth manifold: it is the quotient of the unitary group of the commutant $\{A_0\}'$ divided by the unitary subgroup of the commutant $\{P_0, Q_0\}'$, where $(P_0,Q_0)$ is any fixed pair in $D_{A_0}$. Endowed with a natural reductive structure (a linear connection) and the quotient Finsler metric of the operator norm, it behaves as a classic Riemannian space: any two pairs in $D_{A_0}$ are joined by a geodesic of minimal length. Given a base pair $(P_0,Q_0)$, pairs in an open dense subset of $D_{A_0}$ can be joined to $(P_0,Q_0)$ by a {\it unique} minimal geodesic.

math.FA↗
arXiv2017

Poncaré half-space of a C*-algebra

Let $A$ be a C$*^$-algebra. Given a representation $A\subset B(L)$ in a Hilbert space $L$, the set $G^+\subset A$ of positive invertible elements can be thought as the set of inner products in $L$, related to $A$, which are equivalent to the original inner product. The set $G^+$ has a rich geometry, it is a homogeneous space of the invertible group $G$ of $A$, with an invariant Finsler metric. In the present paper we study the tangent bundle $TG^+$ of $G^+$, as a homogenous Finsler space of a natural group of invertible matrices in $M_2(A)$, identifying $TG^+$ with the {\it Poincaré halfspace} $H$ of $A$, $$ H=\{h\in A: Im(h)\ge 0, Im(h) \hbox{ invertible}\}. $$ We show that $\h\simeq TG^+$ has properties similar to those of a space of non-positive constant curvature.

math.OA↗
arXiv2017

Schmidt decomposable products of projections

We characterize operators $T=PQ$ ($P,Q$ orthogonal projections in a Hilbert space $H$) which have a singular value decomposition. A spatial characterizations is given: this condition occurs if and only if there exist orthonormal bases $\{ψ_n\}$ of $R(P)$ and $\{ξ_n\}$ of $R(Q)$ such that $\langleξ_n,ψ_m\rangle=0$ if $n\ne m$. Also it is shown that this is equivalent to $A=P-Q$ being diagonalizable. Several examples are studied, relating Toeplitz, Hankel and Wiener-Hopf operators to this condition. We also examine the relationship with the differential geometry of the Grassmann manifold of underlying the Hilbert space: if $T=PQ$ has a singular value decomposition, then the generic parts of $P$ and $Q$ are joined by a minimal geodesic with diagonalizable exponent.

math.FA↗
arXiv2017

Uncertainty principle and geometry of the infinite Grassmann manifold

We study the pairs of projections $$ P_If=χ_If ,\ \ Q_Jf= \left(χ_J \hat{f}\right)\check{\ } , \ \ f\in L^2(\mathbb{R}^n), $$ where $I, J\subset \mathbb{R}^n$ are sets of finite Lebesgue measure, $χ_I, χ_J$ denote the corresponding characteristic functions and $\hat{\ } , \check{\ }$ denote the Fourier-Plancherel transformation $L^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n)$ and its inverse. These pairs of projections have been widely studied by several authors in connection with the mathematical formulation of Heisenberg's uncertainty principle. Our study is done from a differential geometric point of view. We apply known results on the Finsler geometry of the Grassmann manifold ${\cal P}({\cal H})$ of a Hilbert space ${\cal H}$ to establish that there exists a unique minimal geodesic of ${\cal P}({\cal H})$, which is a curve of the form $$ δ(t)=e^{itX_{I,J}}P_Ie^{-itX_{I,J}} $$ which joins $P_I$ and $Q_J$ and has length $π/2$. As a consequence we obtain that if $H$ is the logarithm of the Fourier-Plancherel map, then $$ \|[H,P_I]\|\ge π/2. $$ The spectrum of $X_{I,J}$ is denumerable and symmetric with respect to the origin, it has a smallest positive eigenvalue $γ(X_{I,J})$ which satisfies $$ \cos(γ(X_{I,J}))=\|P_IQ_J\|. $$

math.FA↗
arXiv2017

Essentially orthogonal subspaces

We study the set ${\cal C}$ consisting of pairs of orthogonal projections $P,Q$ acting in a Hilbert space ${\cal H}$ such that $PQ$ is a compact operator. These pairs have a rich geometric structure which we describe here. They are parted in three subclasses: ${\cal C}_0$ which consists of pairs where $P$ or $Q$ have finite rank, ${\cal C}_1$ of pairs such that $Q$ lies in the restricted Grassmannian (also called Sato Grassmannian) of the polarization ${\cal H}=N(P)\oplus R(P)$, and ${\cal C}_\infty$. Belonging to this last subclass one has the pairs $$ P_If=χ_If ,\ \ Q_Jf= \left(χ_J \hat{f}\right)\check{\ } , \ \ f\in L^2(\mathbb{R}^n), $$ where $I, J\subset \mathbb{R}^n$ are sets of finite Lebesgue measure, $χ_I, χ_J$ denote the corresponding characteristic functions and $\hat{\ } , \check{\ }$ denote the Fourier-Plancherel transform $L^2(\mathbb{R}^2)\to L^2(\mathbb{R}^2)$ and its inverse. We characterize the connected components of these classes: the components of ${\cal C}_0$ are parametrized by the rank, the components of ${\cal C}_1$ are parametrized by the Fredholm index of the pairs, and ${\cal C}_\infty$ is connected. We show that these subsets are (non complemented) differentiable submanifolds of ${\cal B}({\cal H})\times {\cal B}({\cal H})$.

math.FA↗
arXiv2014

Range additivity, shorted operator and the Sherman-Morrison-Woodbury formula

We say that two operators A, B have the range additivity property if R(A + B) = R(A) + R(B). In this article we study the relationship between range additivity, shorted operator and certain Hilbert space decomposition known as compatibility. As an application, we extend to infinite dimensional Hilbert space operators a formula by Fill and Fishkind related to the well-known Sherman-Morrison-Woodbury formula.

math.FA↗
arXiv2013

Weighted projections into closed subspaces

In this paper we study $A$-projections, i.e. operators of a Hilbert space $\HH$ which act as projections when a seminorm is considered in $\HH$. $A$-projections were introduced by Mitra and Rao \cite{[MitRao74]} for finite dimensional spaces. We relate this concept to the theory of compatibility between positive operators and closed subspaces of $\HH$. We also study the relationship between weighted least squares problems and compatibility.

math.FA↗
arXiv2010

Products of orthogonal projections and polar decompositions

We characterize the sets $\XX$ of all products $PQ$, and $\YY$ of all products $PQP$, where $P,Q$ run over all orthogonal projections and we solve the problems $\arg\min\{\|P-Q\|: (P,Q) \in \cal Z\}$, for $\cal Z=\XX$ or $\YY.$ We also determine the polar decompositions and Moore-Penrose pseudoinverses of elements of $\XX.$

math.FA↗
arXiv2005

Bilateral Shorted Operators and Parallel Sums

In this paper we study shorted operators relative to two different subspaces, for bounded operators on infinite dimensional Hilbert spaces. We define two notions of complementability in the sense of Ando for operators, and study the properties of the shorted operators when they can be defined. We use these facts in order to define and study the notions of parallel sum and substraction, in this Hilbertian context.

math.FA↗
arXiv2005

Metric and homogeneous structure of closed range operators

Let $\CR$ be the set of all bounded linear operators between Hilbert spaces $\cH, \cK$. This paper is devoted to the study of the topological properties of $\CR$ if certain natural metrics are considered on it. We also define an action of the group $\G_\cH\times\G_\cK$ on $\CR$ and determine the orbits of this action. These orbits determine a stratification of the set of Fredholm and semi-Fredholm operators. Finally, we calculate the distance, with respect to some of the metrics mentioned above, between different orbits of $\CR$.

math.FA↗
arXiv2005

Projections in operator ranges

If $\H$ is a Hilbert space, $A$ is a positive bounded linear operator on $\cH$ and $\cS$ is a closed subspace of $\cH$, the relative position between $\cS$ and $A^{-1}(\cS \orto)$ establishes a notion of compatibility. We show that the compatibility of $(A,\cS)$ is equivalent to the existence of a convenient orthogonal projection in the operator range $R(A^{1/2})$ with its canonical Hilbertian structure.

math.FA↗
arXiv2004

Nullspaces and frames

In this paper we give new characterizations of Riesz and conditional Riesz frames in terms of the properties of the nullspace of their synthesis operators. On the other hand, we also study the oblique dual frames whose coefficients in the reconstruction formula minimize different weighted norms.

math.FA↗
arXiv2004

Weighted projections and Riesz frames

Let $\mathcal{H}$ be a (separable) Hilbert space and $\{e_k\}_{k\geq 1}$ a fixed orthonormal basis of $\mathcal{H}$. Motivated by many papers on scaled projections, angles of subspaces and oblique projections, we define and study the notion of compatibility between a subspace and the abelian algebra of diagonal operators in the given basis. This is used to refine previous work on scaled projections, and to obtain a new characterization of Riesz frames.

math.FA↗