Demetrio Stojanoff

Generalized Schur complements and oblique projections

Let S be a closed subspace of a Hilbert space H and A a bounded linear selfadjoint operator on H. In this note, we show that the existence of A-selfadjoint projections with range S is related to some properties of shorted operators, Schur complements (in Ando’s generalization of the classical concept) and compressions.

A note on the star order in Hilbert spaces

We study the star order on the algebra L(ℋ) of bounded operators on a Hilbert space ℋ. We present a new interpretation of this order which allows to generalize to this setting many known results for matrices: functional calculus, semi-lattice properties, shorted operators and orthogonal decompositions. We also show several properties for general Hilbert spaces regarding the star order and its relationship with the functional calculus and the polar decomposition, which were unknown even in the finite-dimensional setting. We also study the existence of strong limits of star-monotone sequences and nets.

Projective spaces of aC*-algebra

Based on the projective matrix spaces studied by B. Schwarz and A. Zaks, we study the notion of projective space associated to a C*-algebraA with a fixed projectionp. The resulting spaceP(p) admits a rich geometrical structure as a holomorphic manifold and a homogeneous reductive space of the invertible group ofA. Moreover, several metrics (chordal, spherical, pseudo-chordal, non-Euclidean-in Schwarz-Zaks terminology) are considered, allowing a comparison amongP(p), the Grassmann manifold ofA 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.

Geometry of the sphere of a Hilbert module

The sphere SX =fx2 X:hx;xi =1 g of a right Hilbert C-module X over a unital C-algebra B is studied using differential geometric techniques. An action of the unitary group of the algebra LB(X) of adjointable B-module operators of X makes SX a homogeneous space of this group. A reductive structure is introduced, as well as a Finsler metric. Metric properties of the geodesic curves are established. In the case B a von Neumann algebra and X self-dual, the fundamental group of SX is computed.

GEOMETRY OF CONDITIONAL EXPECTATIONS AND FINITE INDEX

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

Oblique projections and frames

We characterize those frames on a Hilbert space H which can be represented as the image of an orthonormal basis by an oblique projection defined on an extension K of H. We show that all frames with infinite excess and frame bounds 1 < A < B are of this type. This gives a generalization of a result of Han and Larson which only holds for normalized tight frames.

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.

Index of Hadamard multiplication by positive matrices II

For each n×n positive semidefinite matrix A we define the minimal index I(A)=max{λ⩾0:A∘B⪰λB for all B⪰0} and, for each norm N, the N-index IN(A)=min{N(A∘B):B⪰0 and N(B)=1}, where A∘B=[aijbij] is the Hadamard or Schur product of A=[aij] and B=[bij] and B⪰0 means that B is a positive semidefinite matrix. A comparison between these indexes is done, for different choices of the norm N. As an application we find, for each bounded invertible selfadjoint operator S on a Hilbert space, the best constant M(S) such that ∥STS+S−1TS−1∥⩾M(S)∥T∥ for all T⪰0.

Geometric operator inequalities

Abstract The geometrical meaning of several well-known inequalities is discussed. They include the so-called Loewner, Heinz, McIntosh, and Segal inequalities. It is shown that some of them can be deduced from the others, even for unitarily invariant forms. Some spectral properties of the elementary operators associated to the inequalities are studied.

PROJECTIVE SPACE OF A C*-MODULE

Let X be a right Hilbert C*-module over A. We study the geometry and the topology of the projective space 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 Sp(X) and the natural fibration , where Sp(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 ℒA(X) of adjointable operators of X. The homotopy theory of these spaces is examined.