Raúl E. Curto

The truncated complex -moment problem

Let γ ≡ γ(2n) denote a sequence of complex numbers γ00, γ01, γ10, . . . , γ0,2n, . . . , γ2n,0 (γ00 > 0, γij = γji), and let K denote a closed subset of the complex plane C. The Truncated Complex K-Moment Problem for γ entails determining whether there exists a positive Borel measure μ on C such that γij = ∫ zizj dμ (0 ≤ i + j ≤ 2n) and suppμ ⊆ K. For K ≡ KP a semi-algebraic set determined by a collection of complex polynomials P = {pi (z, z)}mi=1, we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix M (n) (γ) and the localizing matrices Mpi . We prove that there exists a rankM (n)-atomic representing measure for γ(2n) supported in KP if and only if M (n) ≥ 0 and there is some rankpreserving extension M (n+ 1) for which Mpi (n+ ki) ≥ 0, where deg pi = 2ki or 2ki − 1 (1 ≤ i ≤ m).

Solution of the truncated complex moment problem for flat data

Introduction Moment matrices Positive moment matrices and representing measures Existence of representing measures Extension of flat positive moment matrices Applications Generalizations to several variables References List of symbols.

Hyponormal Pairs of Commuting Operators

We analyze the notions of weak and strong joint hyponormality for commuting pairs of operators, with an aim at understanding the gap between hyponormality and subnormality for single operators. We exhibit a commuting pair T = (T1, T2) such that: (i) T is weakly hyponormal; (ii) T is not strongly hyponormal; (iii) T 1 l 1T 2 l 2 is subnormal (all l1, l2 ≥ 0); (iv) T1 + T2 is not subnormal; (v) T1 + T2 is power hyponormal; and (vi) T1 is unitarily equivalent to T2.

Flat extensions of positive moment matrices: recursively generated relations

Introduction Flat extensions for moment matrices The singular quartic moment problem The algebraic variety of

Joint hyponormality of Toeplitz pairs

\gamma

WEYL'S THEOREM,

J. E. McCarthys phenomenon and the proof of Theorem 1.5 Summary of results Bibliography List of symbols.

a

Introduction Hyponormality of Toeplitz pairs with one coordinate a Toeplitz operator with analytic polynomial symbol Hyponormality of trigonometric Toeplitz pairs The gap between

-WEYL'S THEOREM, AND LOCAL SPECTRAL THEORY

2

Flat Extensions of Positive Moment Matrices: Relations in Analytic or Conjugate Terms

-hyponormality and subnormality Applications Concluding remarks and open problems References List of symbols.

Algebraic methods in operator theory

We give necessary and sufficient conditions for a Banach space operator with the single valued extension property (SVEP) to satisfy Weyl’s theorem and a-Weyl’s theorem. We show that if T or T ∗ has SVEP and T is transaloid, then Weyl’s theorem holds for f(T ) for every f ∈ H(σ(T )). When T ∗ has SVEP, T is transaloid and T is a-isoloid, then a-Weyl’s theorem holds for f(T ) for every f ∈ H(σ(T )). We also prove that if T or T ∗ has SVEP, then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.