# Numerical Range and Quadratic Numerical Range for Damped Systems

## TL;DR

This paper develops new spectral enclosures for non-selfadjoint operator matrices in damped systems using quadratic numerical range, providing tighter bounds and explicit estimates that improve upon previous results.

## Contribution

It introduces novel spectral bounds for damped systems leveraging quadratic numerical range, applicable under weak assumptions on damping operators.

## Key findings

- Spectral enclosures can have bounded imaginary parts.
- New bounds improve earlier sectorial and selfadjoint D results.
- Explicit bounds demonstrated in fluid flow example.

## Abstract

We prove new enclosures for the spectrum of non-selfadjoint operator matrices associated with second order linear differential equations $\ddot{z}(t) + D \dot{z} (t) + A_0 z(t) = 0$ in a Hilbert space. Our main tool is the quadratic numerical range for which we establish the spectral inclusion property under weak assumptions on the operators involved; in particular, the damping operator only needs to be accretive and may have the same strength as $A_0$. By means of the quadratic numerical range, we establish tight spectral estimates in terms of the unbounded operator coefficients $A_0$ and $D$ which improve earlier results for sectorial and selfadjoint $D$; in contrast to numerical range bounds, our enclosures may even provide bounded imaginary part of the spectrum or a spectral free vertical strip. An application to small transverse oscillations of a horizontal pipe carrying a steady-state flow of an ideal incompressible fluid illustrates that our new bounds are explicit.

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Source: https://tomesphere.com/paper/1703.07447