Asao Arai
Operator Theory on a Commutation Relation
(59K, Latex2.09)
ABSTRACT. An operator theory is developed
for a pair $(A,H)$ with $A$ being a symmetric operator
on a Hilbert space ${\cal H}$ and $H$ an injective
self-adjoint
operator on ${\cal H}$ such that
$D(H^2)\subset D(A)$ (for a linear
operator $T$, $D(T)$ denotes its domain) and
$\lang H^2\psi,A\phi\rang=\lang A\psi, H^2\phi\rang,
\, \psi,\phi \in D(H^2)$, where
$\lang\,\cdot\,,\,\cdot\,\rang$ is the inner product of ${\cal H}$.
One of the main results of the present paper
is concerned with a decomposition theorem
of $\bar A$ (the closure of $A$)\,: Under suitable conditions,
there exists a unique pair $(A_+,A_-)$ of
self-adjoint operators such that the following hold\,:
(i) $\bar A=A_++A_-$\,;
(ii) $A_+$ strongly commutes with $H$\,; (iii) $A_-$
strongly anticommutes with $H$.
Moreover, under some additional
condition, $A_+$ strongly anticommutes with $A_-$.
It is shown that, under suitable conditions, the spectrum
of $H$ is symmetric with respect to the origin in $\R$.
As an application, we define a class of
linear operators $X$ on ${\cal H}$ for which $e^{itH}Xe^{-itH}$
($t\in \R$) has
an explicit representation. A class of pairs $(A,H)$
to which the general theory can be applied
is given with $H$ being an abstract form of
Dirac type operators.