1. CMB 2008 (vol 51 pp. 86)
|The Numerical Range of 2-Dimensional Krein Space Operators |
The tracial numerical range of operators on a $2$-dimensional Krein space is investigated. Results in the vein of those obtained in the context of Hilbert spaces are obtained.
Keywords:numerical range, generalized numerical range, indefinite inner product space
Categories:15A60, 15A63, 15A45
2. CMB 2005 (vol 48 pp. 394)
|Diagonal Plus Tridiagonal Representatives for Symplectic Congruence Classes of Symmetric Matrices |
Let $n=2m$ be even and denote by $\Sp_n(F)$ the symplectic group of rank $m$ over an infinite field $F$ of characteristic different from $2$. We show that any $n\times n$ symmetric matrix $A$ is equivalent under symplectic congruence transformations to the direct sum of $m\times m$ matrices $B$ and $C$, with $B$ diagonal and $C$ tridiagonal. Since the $\Sp_n(F)$-module of symmetric $n\times n$ matrices over $F$ is isomorphic to the adjoint module $\sp_n(F)$, we infer that any adjoint orbit of $\Sp_n(F)$ in $\sp_n(F)$ has a representative in the sum of $3m-1$ root spaces, which we explicitly determine.
Categories:11E39, 15A63, 17B20
3. CMB 2004 (vol 47 pp. 73)
|Systems of Hermitian Quadratic Forms |
In this paper, we give some conditions to judge when a system of Hermitian quadratic forms has a real linear combination which is positive definite or positive semi-definite. We also study some related geometric and topological properties of the moduli space.
Keywords:hermitian quadratic form, positive definite, positive semi-definite