Expand all Collapse all | Results 1 - 3 of 3 |
1. CMB 2004 (vol 47 pp. 369)
Spectrally Bounded Linear Maps on ${\cal B}(X)$ We characterize surjective linear maps on ${\cal B}(X)$ that are
spectrally bounded and spectrally bounded below.
Keywords:spectrally bounded linear map. Category:47B49 |
2. CMB 2003 (vol 46 pp. 216)
Linear Maps on Selfadjoint Operators Preserving Invertibility, Positive Definiteness, Numerical Range |
Linear Maps on Selfadjoint Operators Preserving Invertibility, Positive Definiteness, Numerical Range Let $H$ be a complex Hilbert space, and $\HH$ be the real linear space of
bounded selfadjoint operators on $H$. We study linear maps $\phi\colon \HH
\to \HH$ leaving invariant various properties such as invertibility, positive
definiteness, numerical range, {\it etc}. The maps $\phi$ are not assumed
{\it a priori\/} continuous. It is shown that under an appropriate surjective
or injective assumption $\phi$ has the form $X \mapsto \xi TXT^*$ or $X \mapsto
\xi TX^tT^*$, for a suitable invertible or unitary $T$ and $\xi\in\{1, -1\}$,
where $X^t$ stands for the transpose of $X$ relative to some orthonormal basis.
Examples are given to show that the surjective or injective assumption cannot
be relaxed. The results are extended to complex linear maps on the algebra of
bounded linear operators on $H$. Similar results are proved for the (real)
linear space of (selfadjoint) operators of the form $\alpha I+K$, where $\alpha$
is a scalar and $K$ is compact.
Keywords:linear map, selfadjoint operator, invertible, positive definite, numerical range Categories:47B15, 47B49 |
3. CMB 2003 (vol 46 pp. 54)
Linear Maps Transforming the Unitary Group Let $U(n)$ be the group of $n\times n$ unitary matrices. We show that if
$\phi$ is a linear transformation sending $U(n)$ into $U(m)$, then $m$ is
a multiple of $n$, and $\phi$ has the form
$$
A \mapsto V[(A\otimes I_s)\oplus (A^t \otimes I_{r})]W
$$
for some $V, W \in U(m)$. From this result, one easily deduces the
characterization of linear operators that map $U(n)$ into itself obtained
by Marcus. Further generalization of the main theorem is also discussed.
Keywords:linear map, unitary group, general linear group Category:15A04 |