Numerical shadows: measures and densities on the numerical range

TL;DR

This paper introduces the concept of numerical shadows for operators on finite-dimensional Hilbert spaces, analyzing their geometric and probabilistic properties, explicit forms, and connections to higher-rank numerical ranges.

Contribution

It defines numerical shadows for operators, provides explicit descriptions for special cases, and links these shadows to geometric interpretations and higher-rank numerical ranges.

Findings

Numerical shadow of a hermitian operator is a B-spline distribution.

Normal operators' shadows correspond to projections of simplices.

Explicit formulas for Jordan matrices and rotation-invariant cases.

Abstract

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.