Numerical shadows: measures and densities on the numerical range

For any operator $M$ acting on an $N$-dimensional Hilbert space $H_N$ we introduce its numerical shadow, which is a probability measure on the complex plane supported by the numerical range of $M$. The shadow of $M$ at point $z$ is defined as the probability that the inner product $(Mu,u)$ is equal to $z$, where $u$ stands for a random complex vector from $H_N$, satisfying $||u||=1$. In the case of N=2 the numerical shadow of a non-normal operator can be interpreted as a shadow of a hollow sphere projected on a plane. A similar interpretation is provided also for higher dimensions. For a hermitian $M$ its numerical shadow forms a probability distribution on the real axis which is shown to be a one dimensional $B$-spline. In the case of a normal $M$ the numerical shadow corresponds to a shadow of a transparent solid simplex in $R^{N-1}$ onto the complex plane. Numerical shadow is found explicitly for Jordan matrices $J_N$, direct sums of matrices and in all cases where the shadow is rotation invariant. Results concerning the moments of shadow measures play an important role. A general technique to study numerical shadow via the Cartesian decomposition is described, and a link of the numerical shadow of an operator to its higher-rank numerical range is emphasized.Comment: 37 pages, 8 figure

Quaternion Matrices : Statistical Properties and Applications to Signal Processing and Wavelets

Similarly to how complex numbers provide a possible framework for extending scalar signal processing techniques to 2-channel signals, the 4-dimensional hypercomplex algebra of quaternions can be used to represent signals with 3 or 4 components. For a quaternion random vector to be suited for quaternion linear processing, it must be (second-order) proper. We consider the likelihood ratio test (LRT) for propriety, and compute the exact distribution for statistics of Box type, which include this LRT. Various approximate distributions are compared. The Wishart distribution of a quaternion sample covariance matrix is derived from first principles. Quaternions are isomorphic to an algebra of structured 4x4 real matrices. This mapping is our main tool, and suggests considering more general real matrix problems as a way of investigating quaternion linear algorithms. A quaternion vector autoregressive (VAR) time-series model is equivalent to a structured real VAR model. We show that generalised least squares (and Gaussian maximum likelihood) estimation of the parameters reduces to ordinary least squares, but only if the innovations are proper. A LRT is suggested to simultaneously test for quaternion structure in the regression coefficients and innovation covariance. Matrix-valued wavelets (MVWs) are generalised (multi)wavelets for vector-valued signals. Quaternion wavelets are equivalent to structured MVWs. Taking into account orthogonal similarity, all MVWs can be constructed from non-trivial MVWs. We show that there are no non-scalar non-trivial MVWs with short support [0,3]. Through symbolic computation we construct the families of shortest non-trivial 2x2 Daubechies MVWs and quaternion Daubechies wavelets.Open Acces

Numerical Computation of Resonances and Pseudospectra in Acoustic Scattering

Acoustic scattering is a well-known physical phenomenon which arises in a wide range of fields: when acoustic waves propagating in a medium impinge on a localised non-uniformity, such as a density fluctuation or an external obstacle, their trajectories are deviated and scattered waves are generated. A key role in scattering theory is played by resonances; these are particular scatterer-dependent non-physical ‘complex’ frequencies at which acoustic scattering exhibits exceptional behaviour. The study of acoustic resonances for a particular scatterer provides an insight in the behaviour that the acoustic scattering assumes at the near physical ‘real’ frequencies, and it is a fundamental step in many applications. Yet, the numerical computation of resonances and pseudospectra - a mathematical tool which can be used to study the influence of resonances on physical frequencies - remains very expensive. With the present Thesis we want to address this particular problem, by proposing numerical algorithms based on the Boundary Element Method (BEM) for computing resonances and pseudospectra and by analysing their efficiency and performance. Finally, we apply such algorithms to half a dozen of physically relevant scatterer, inspired from different fields where acoustic scattering plays a relevant role

K-spectral sets, operator tuples and related function theory

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 28-04-2017La investigación en la que se basa esta tesis ha sido financiada por los proyectos MTM2011-28149-C02-1 y MTM2015-66157-C2-1-P del Ministerio de Economía y Competitividad, cuyo investigador principal es José Luis Torrea