The normalized numerical range and the Davis-Wielandt shell
Abstract
For a given $n$-by-$n$ matrix $A$, its {\em normalized numerical range} $F_N(A)$ is defined as the range of the function $f_{N,A}\colon x\mapsto (x^*Ax)/(\norm{Ax}\cdot\norm{x})$ on the complement of $\ker A$. We provide an explicit description of this set for the case when $A$ is normal or $n=2$. This extension of earlier results for particular cases of $2$-by-$2$ matrices (by Gevorgyan) and essentially Hermitian matrices of arbitrary size (by A. Stoica and one of the authors) was achieved due to the fresh point of view at $F_N(A)$ as the image of the Davis-Wielandt shell $\JNR(A)$ under a certain non-linear mapping $h\colon\R^3\mapsto\C$.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2017
- DOI:
- 10.48550/arXiv.1712.01070
- arXiv:
- arXiv:1712.01070
- Bibcode:
- 2017arXiv171201070L
- Keywords:
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- Mathematics - Functional Analysis;
- Mathematics - Dynamical Systems;
- 15A60;
- 47A12;
- 47B15
- E-Print:
- 23 pages, 4 figures