esteban andruchow - Academia.edu (original) (raw)

Papers by esteban andruchow

arXiv (Cornell University), Jul 18, 2019

We study pairs (U, L 0), where U is a unitary operator in H and L 0 ⊂ H is a closed subspace, suc... more We study pairs (U, L 0), where U is a unitary operator in H and L 0 ⊂ H is a closed subspace, such that P L0 U | L0 : L 0 → 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.

arXiv (Cornell University), Apr 2, 2020

Let C(H) = B(H)/K(H) be the Calkin algebra (B(H) the algebra of bounded operators on the Hilbert ... more Let C(H) = B(H)/K(H) be the Calkin algebra (B(H) the algebra of bounded operators on the Hilbert space H, K(H) the ideal of compact operators, and π : B(H) → C(H) the quotient map), and P C(H) the differentiable manifold of selfadjoint projections in C(H). A projection p in C(H) can be lifted to a projection P ∈ B(H): π(P) = p. We show that given p, q ∈ P C(H) , there exists a minimal geodesic of P C(H) which joins p and q if and only if there exist lifting projections P and Q such that either both N (P − Q ± 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) ∈ P C(H) , t ∈ I, joining the same endpoints, where the length of γ is measured by I γ(t) dt.

arXiv (Cornell University), Dec 9, 2021

A known general program, designed to endow the quotient space U A /U B of the unitary groups U A ... more A known general program, designed to endow the quotient space U A /U B of the unitary groups U A , U B of the C * algebras B ⊂ A with an invariant Finsler metric, is applied to obtain a metric for the space I(H) of partial isometries of a Hilbert space H. I(H) is a quotient of the unitary group of B(H) × B(H), where B(H) is the algebra of bounded linear operators in 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.

arXiv (Cornell University), May 17, 2018

The set D A0 , of pairs of orthogonal projections (P, Q) in generic position with fixed differenc... more The set D A0 , 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 A0. 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 A0 are joined by a geodesic of minimal length. Given a base pair (P 0 , Q 0), pairs in an open dense subset of D A0 can be joined to (P 0 , Q 0) by a unique minimal geodesic.

arXiv (Cornell University), Mar 14, 2019

We study the elementary C *-algebra D + K which consists of sums of a diagonal plus a compact ope... more We study the elementary C *-algebra D + K which consists of sums of a diagonal plus a compact operator. We describe the structure of the unitary group, the sets of ideals, automorhisms and projections.

Mathematische Nachrichten, Apr 19, 2023

Let X be a right C*‐module over a unital C*‐algebra . We study the Hopf fibration of X: where the... more Let X be a right C*‐module over a unital C*‐algebra . We study the Hopf fibration of X: where the projective space of X is the set of singly generated orthocomplemented submodules of X, is the set of elements of X, which generate such submodules, and module generated by . The group of unitary operators of the module X acts on both spaces. We introduce a Finsler metric in , which is invariant under the unitary action. Our main results establish that the map is distance decreasing (when the projective space of X is considered with its natural unitary invariant metric), and a minimality result in , characterizing metric geodesics in this space.

Acta Scientiarum Mathematicarum, Aug 1, 2022

arXiv (Cornell University), Mar 2, 2018

For a given Hilbert space H, consider the space of self-adjoint projections P(H). In this paper w... more For a given Hilbert space H, consider the space of self-adjoint projections P(H). In this paper we study the differentiable structure of a canonical sphere bundle over P(H) given by R = { (P, f) ∈ P(H) × H : P f = f, f = 1 }. We establish the smooth action on R of the group of unitary operators of H, therefore R is an homogeneous space. Then we study the metric structure of R by endowing it first with the uniform quotient metric, which is a Finsler metric, and we establish minimality results for the geodesics. These are given by certain one-parameter groups of unitary operators, pushed into R by the natural action of the unitary group. Then we study the restricted bundle R + 2 given by considering only the projections in the restricted Grassmannian, locally modelled by Hilbert-Schmidt operators. Therefore we endow R + 2 with a natural Riemannian metric that can be obtained by declaring that the action of the group is a Riemannian submersion. We study the Levi-Civita connection of this metric and establish a Hopf-Rinow theorem for R + 2 , again obtaining a characterization of the geodesics as the image of certain one-parameter groups with special speeds. 1

arXiv (Cornell University), Jul 31, 2020

Let H be a reproducing kernel Hilbert space of functions on a set X. We study the problem of find... more Let 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 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.

arXiv (Cornell University), Jul 3, 2023

We study the composition operators C a acting on the Hardy space H 2 of the unit disk, given by C... more We study the composition operators C a acting on the Hardy space H 2 of the unit disk, given by C a f = f • φ a , where φ a (z) = a − z 1 −āz , for |a| < 1. These operators are reflections: C 2 a = 1. We study their eigenspaces N (C a ± 1), their relative position (i.e., the intersections between these spaces and their orthogonal complementes for a ̸ = b in the unit disk) and the symmetries induced by C a and these eigenspaces.

arXiv (Cornell University), Jun 15, 2017

We characterize operators T = P Q (P, Q orthogonal projections in a Hilbert space H) which have a... more We characterize operators T = P Q (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 ξ n , ψ m = 0 if n = 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 = P Q has a singular value decomposition, then the generic parts of P and Q are joined by a minimal geodesic with diagonalizable exponent.

arXiv (Cornell University), Aug 19, 2016

Let L 2 be the Lebesgue space of square-integrable functions on the unit circle. We show that the... more Let L 2 be the Lebesgue space of square-integrable functions on the unit circle. We show that the injectivity problem for Toeplitz operators is linked to the existence of geodesics in the Grassmann manifold of L 2. We also investigate this connection in the context of restricted Grassmann manifolds associated to p-Schatten ideals and essentially commuting projections. 1

International Journal of Mathematics, Apr 1, 1994

Let e be the Jones projection associated to a conditional expectation [Formula: see text] where [... more Let e be the Jones projection associated to a conditional expectation [Formula: see text] where [Formula: see text] are von Neumann algebras. We prove that the similarity orbit of e by invertibles of [Formula: see text] is an homogeneous space iff the index of E is finite. If also [Formula: see text], then this orbit is a covering space for the orbit of E.

arXiv (Cornell University), Jul 18, 2023

Let Gr be a component of the Grassmann manifold of a C *-algebra, presented as the unitary orbit ... more 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 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 ∈ Gr for the spectral rectifiable distance, and also the conjugate tangent locus of P ∈ 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. Contents 1. Introduction 2.2. Linear connections 2.3. Levi-Civita connection of the trace 2.4. Complex structure and symplectic form 2.5. Symmetric space structure 2.6. Jacobi fields 3. Cut locus and conjugate locus 3.1. Cut locus 3.2. Conjugate points 3.3. The kernel of D Exp P 3.4. Projective spaces 3.5. Beyond first conjugate point References

arXiv (Cornell University), Nov 20, 1999

Let X be a right Hilbert C *-module over A. We study the geometry and the topology of the project... more Let X be a right Hilbert C *-module over A. We study the geometry and the topology of the projective space P(X) of X , consisting of the orthocomplemented submodules of X which are generated by a single element. We also study the geometry of the p-sphere S p (X) and the natural fibration S p (X) → P(X) , where S p (X) = {x ∈ X :< x, x >= p} , for p ∈ A a projection. The projective space and the p-sphere are shown to be homogeneous differentiable spaces of the unitary group of the algebra L A (X) of adjointable operators of X. The homotopy theory of these spaces is examined.

arXiv (Cornell University), Nov 18, 1999

Based on the projective matrix spaces studied by B. Schwarz and A. Zaks, we study the notion of p... more Based on the projective matrix spaces studied by B. Schwarz and A. Zaks, we study the notion of projective space associated to a C *-algebra A with a fixed projection p. The resulting space P(p) admits a rich geometrical structure as a holomorphic manifold and a homogeneous reductive space of the invertible group of A. Moreover, several metrics (chordal, spherical, pseudo-chordal, non-Euclidean-in Schwarz-Zaks terminology) are considered, allowing a comparison among P(p), the Grassmann manifold of A and the space of positive elements which are unitary with respect to the bilinear form induced by the reflection ε = 2p − 1. Among several metrical results, we prove that geodesics are unique and of minimal length when measured with the spherical and non-Euclidean metrics.

arXiv (Cornell University), Oct 5, 2010

Let A be a von Neumann algebra with a finite trace τ , represented in H = L 2 (A, τ), and let Bt ... more Let A be a von Neumann algebra with a finite trace τ , represented in H = L 2 (A, τ), and let Bt ⊂ A be sub-algebras, for t in an interval I (0 ∈ I). Let Et : A → Bt be the unique τ-preserving conditional expectation. We say that the path t → Et is smooth if for every a ∈ A and ξ ∈ H, the map I ∋ t → Et(a)ξ ∈ H is continuously differentiable. This condition implies the existence of the derivative operator dEt(a) : H → H, dEt(a)ξ = d dt Et(a)ξ. If this operator verifies the additional boundedness condition, J dEt(a) 2 2 dt ≤ CJ a 2 2 , for any closed bounded sub-interval J ⊂ I, and CJ > 0 a constant depending only on J, then the algebras Bt are *-isomorphic. More precisely, there exists a curve Gt : A → A, t ∈ I of unital, *-preserving linear isomorphisms which intertwine the expectations, Gt • E0 = Et • Gt. The curve Gt is weakly continuously differentiable. Moreover, the intertwining property in particular implies that Gt maps B0 onto Bt. We show that this restriction is a multiplicative isomorphism. 1

Proceedings of the American Mathematical Society, 1999

It is proven that the Löwner-Heinz inequality ‖ A t B t ‖ ≤ ‖ A B ‖ t {\|A^{t}B^{t}\|\le \|AB\|^{... more It is proven that the Löwner-Heinz inequality ‖ A t B t ‖ ≤ ‖ A B ‖ t {\|A^{t}B^{t}\|\le \|AB\|^{t}} , valid for all positive invertible operators A , B {A, B} on the Hilbert space H {\mathcal H } and t ∈ [ 0 , 1 ] {t\in [0,1]} , has equivalent forms related to the Finsler structure of the space of positive invertible elements of L ( H ) {\mathcal L (\mathcal H )} or, more generally, of a unital C ∗ {C^{*}} -algebra. In particular, the Löwner-Heinz inequality is equivalent to some type of “nonpositive curvature" property of that space.

arXiv (Cornell University), Jan 13, 2017

We study the pairs of projections P I f = χ I f, Q J f = χ Jf ˇ, f ∈ L 2 (R n), where I, J ⊂ R n ... more We study the pairs of projections P I f = χ I f, Q J f = χ Jf ˇ, f ∈ L 2 (R n), where I, J ⊂ R n are sets of finite Lebesgue measure, χ I , χ J denote the corresponding characteristic functions andˆ,ˇdenote the Fourier-Plancherel transformation L 2 (R n) → L 2 (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 P(H) of a Hilbert space H to establish that there exists a unique minimal geodesic of P(H), which is a curve of the form δ(t) = e itXI,J P I e −itXI,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 ] ≥ 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 I Q J .

Complex Analysis and Operator Theory, 2016

We study the sets of idempotents that one obtains when the entry E_{1,2} is a special type of ope... more We study the sets of idempotents that one obtains when the entry E_{1,2} is a special type of operator: compact, Fredholm and injective with dense range, among others.