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

How wages change: micro evidence from the International Wage Flexibility Project

How do the complex institutions involved in wage setting affect wage changes? The International Wage Flexibility Project provides new microeconomic evidence on how wages change for continuing workers. We analyze individuals’ earnings in 31 different data sets from sixteen countries, from which we obtain a total of 360 wage change distributions. We find a remarkable amount of variation in wage changes across workers. Wage changes have a notably non-normal distribution; they are tightly clustered around the median and also have many extreme values. Furthermore, nearly all countries show asymmetry in their wage distributions below the median. Indeed, we find evidence of both downward nominal and real wage rigidities. We also find that the extent of both these rigidities varies substantially across countries. Our results suggest that variations in the extent of union presence in wage bargaining play a role in explaining differing degrees of rigidities among countries. JEL Classification: E3, J3, J5Downward nominal wage rigidity, Downward real wage rigidity, Wage change distributions, Wage setting