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1. CJM 2009 (vol 62 pp. 646)
Reducibility in A_{R}(K), C_{R}(K), and A(K) Let $K$ denote a compact real symmetric subset of $\mathbb{C}$ and let
$A_{\mathbb R}(K)$ denote the real Banach algebra of all real
symmetric continuous functions on $K$ that are analytic in the
interior $K^\circ$ of $K$, endowed with the supremum norm. We
characterize all unimodular pairs $(f,g)$ in $A_{\mathbb R}(K)^2$
which are reducible.
In addition, for an arbitrary compact $K$ in $\mathbb C$, we give a
new proof (not relying on Banach algebra theory or elementary stable
rank techniques) of the fact that the Bass stable rank of $A(K)$ is
$1$.
Finally, we also characterize all compact real symmetric sets $K$ such
that $A_{\mathbb R}(K)$, respectively $C_{\mathbb R}(K)$, has Bass
stable rank $1$.
Keywords:real Banach algebras, Bass stable rank, topological stable rank, reducibility Categories:46J15, 19B10, 30H05, 93D15 |
2. CJM 1997 (vol 49 pp. 840)
Non-Hermitian solutions of algebraic Riccati equation Non-hermitian solutions of algebraic matrix Riccati
equations (of the continuous and discrete types) are studied. Existence
is proved of non-hermitian solutions with given upper bounds of the
ranks of the skew-hermitian parts, under the sign controllability
hypothesis.
Categories:15A99, 15A63, 93C60 |