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.