Quadratic Cones Invariant under Some Linear Operators
Dragomir Ž. Ðoković · SIAM Journal on Algebraic and Discrete Methods · 1987
A (solid) quadratic cone K in a finite-dimensional vector space V (over ${\bf R}, {\bf C}$, or ${\bf H}$) is the set of all $x \in V$ satisfying $f(x,x)\geqq 0$, where f is a fixed indefinite hermitian form. Given such a cone K, we characterize the linear operators A for which $AK \subset K$, and also those for which $AK = K$. We also show that if $\rho (A) = u (A)$ for some (multiplicative) norm $ u$ on the algebra of linear operators ($\rho $ denotes the spectral radius) then there exists an A-invariant quadratic cone of specified signature. For this purpose we strengthen a result of Mott and Schneider characterizing the operators A for which $\rho (A) = u (A)$ is possible.