We consider a non-self-adjoint h-pseudodifferential operator P in the semi-classical limit (h → 0). If p is the leading symbol, then under suitable assumptions about the behaviour of p at infinity, we know that the resolvent (z − P )−1 is uniformly bounded for z in any compact set not intersecting the closure of the range of p. Under a subellipticity condition, we show that the resolvent extend...

Maximal and atomic Hardy spaces Hp and H A, 0 < p ≤ 1, are considered in the setting of a doubling metric measure space in the presence of a non-negative self-adjoint operator whose heat kernel has Gaussian localization and the Markov property. It is shown that Hp = H A with equivalent norms.

The eigenvalues of a self-adjoint n×n matrix A can be put into a decreasing sequence λ = (λ1, . . . , λn), with repetitions according to multiplicity, and the diagonal of A is a point of R that bears some relation to λ. The Schur-Horn theorem characterizes that relation in terms of a system of linear inequalities. We prove an extension of the latter result for positive trace-class operators on ...

We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from self-adjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the ...

Maximal and atomic Hardy spaces Hp and H A, 0 < p ≤ 1, are considered in the setting of a doubling metric measure space in the presence of a non-negative self-adjoint operator whose heat kernel has Gaussian localization and the Markov property. It is shown that Hp = H A with equivalent norms.

Mini course 4×50 minutes, in Rennes, 14-17 June 2011. Résumé Mini-cours 4×50 minutes à Rennes, 14-17 juin 2011.

Self-adjoint operators and their spectra play a crucial rôle in analysis and physics. For instance, in quantum physics self-adjoint operators are used to describe measurements and the spectrum represents the set of possible measurement results. Therefore, it is a natural question whether the spectrum of a self-adjoint operator can be computed from a description of the operator. We prove that gi...