Polar decomposition under perturbations of the scalar product.
Operator inequalities, geodesics and interpolation
Extension of characters in commutative Banach algebras
Total curvature of non-differentiable curves.
Metrics in the set of partial isometries with finite rank
Let be the set of partial isometries with finite rank of an infinite dimensional Hilbert space . We show that is a smooth submanifold of the Hilbert space of Hilbert-Schmidt operators of and that each connected component is the set , which consists of all partial isometries of rank . Furthermore, is a homogeneous space of , where is the classical Banach-Lie group of unitary operators of , which are Hilbert-Schmidt perturbations of the identity. We introduce two Riemannian metrics...
Geometry of oblique projections
Let A be a unital C*-algebra. Denote by P the space of selfadjoint projections of A. We study the relationship between P and the spaces of projections determined by the different involutions induced by positive invertible elements a ∈ A. The maps sending p to the unique with the same range as p and sending q to the unitary part of the polar decomposition of the symmetry 2q-1 are shown to be diffeomorphisms. We characterize the pairs of idempotents q,r ∈ A with ||q-r|| < 1 such that...
A note on the differentiable structure of generalized idempotents
For a fixed n > 2, we study the set Λ of generalized idempotents, which are operators satisfying T n+1 = T. Also the subsets Λ†, of operators such that T n−1 is the Moore-Penrose pseudo-inverse of T, and Λ*, of operators such that T n−1 = T* (known as generalized projections) are studied. The local smooth structure of these sets is examined.
A classification of projectors
A positive operator A and a closed subspace of a Hilbert space ℋ are called compatible if there exists a projector Q onto such that AQ = Q*A. Compatibility is shown to depend on the existence of certain decompositions of ℋ and the ranges of A and . It also depends on a certain angle between A() and the orthogonal of .
Characterization of Bessel sequences.
Let H be a separable Hilbert space, L(H) be the algebra of all bounded linear operators of H and (H) be the set of all Bessel sequences of H. Fixed an orthonormal basis E = {e} of H, a bijection α: (H) → L(H) can be defined. The aim of this paper is to characterize α (A) for different classes of operators A ⊆ L(H). In particular, we characterize the Bessel sequences associated to injective operators, compact operators and Schatten p-classes.
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