Numerical shadows: Measures and densities on the numerical range

For any operator M acting on an N-dimensional Hilbert space HN 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 HN, 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 RN-1 onto the complex plane. Numerical shadow is found explicitly for Jordan matrices JN, 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.