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1. CMB 2011 (vol 54 pp. 456)
On Operator Sum and Product Adjoints and Closures We comment on domain conditions that regulate when the adjoint of the
sum or product of two unbounded operators is the sum or product of their
adjoints, and related closure issues. The quantum mechanical problem PHP
essentially selfadjoint for unbounded Hamiltonians is addressed, with new
results.
Keywords:unbounded operators, adjoints of sums and products, quantum mechanics Category:47A05 |
2. CMB 2011 (vol 54 pp. 498)
On the Adjoint and the Closure of the Sum of Two Unbounded Operators We prove, under some conditions on the domains, that the adjoint of
the sum of two unbounded operators is the sum of their adjoints in
both Hilbert and Banach space settings. A similar result about the
closure of operators is also proved. Some interesting consequences
and examples "spice up" the paper.
Keywords:unbounded operators, sum and products of operators, Hilbert and Banach adjoints, self-adjoint operators, closed operators, closure of operators Category:47A05 |
3. CMB 2010 (vol 54 pp. 21)
Generalized D-symmetric Operators II
Let $H$ be a separable,
infinite-dimensional, complex Hilbert space and let $A, B\in{\mathcal L
}(H)$, where ${\mathcal L}(H)$ is the algebra of all bounded linear
operators on $H$. Let $\delta_{AB}\colon {\mathcal L}(H)\rightarrow {\mathcal
L}(H)$ denote the generalized derivation $\delta_{AB}(X)=AX-XB$.
This note will initiate a study on the class of pairs $(A,B)$ such
that $\overline{{\mathcal R}(\delta_{AB})}= \overline{{\mathcal
R}(\delta_{A^{\ast}B^{\ast}})}$.
Keywords:generalized derivation, adjoint, D-symmetric operator, normal operator Categories:47B47, 47B10, 47A30 |
4. CMB 2006 (vol 49 pp. 55)
Non Abelian Twisted Reidemeister Torsion for Fibered Knots In this article, we give an explicit formula to compute the
non abelian twisted sign-deter\-mined Reidemeister torsion of the
exterior of a fibered knot in terms of its monodromy. As an
application, we give explicit formulae for the non abelian
Reidemeister torsion of torus knots and of the figure eight knot.
Keywords:Reidemeister torsion, Fibered knots, Knot groups, Representation space, $\SU$, $\SL$, Adjoint representation, Monodromy Categories:57Q10, 57M27, 57M25 |
5. 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 |
6. CMB 1998 (vol 41 pp. 267)
On the nonemptiness of the adjoint linear system of polarized manifold Let $(X,L)$ be a polarized manifold over the complex number field
with $\dim X=n$. In this paper, we consider a conjecture of
M.~C.~Beltrametti and A.~J.~Sommese and we obtain that this
conjecture is true if $n=3$ and $h^{0}(L)\geq 2$, or $\dim \Bs
|L|\leq 0$ for any $n\geq 3$. Moreover we can generalize the
result of Sommese.
Keywords:Polarized manifold, adjoint bundle Categories:14C20, 14J99 |