Gustavo Corach

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.

Projections in operator ranges

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

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 Tn+1 = T. Also the subsets Λ†, of operators such that Tn−1 is the Moore-Penrose pseudo-inverse of T, and Λ*, of operators such that Tn−1 = T* (known as generalized projections) are studied. The local smooth structure of these sets is examined.