06-36 Asao Arai
Operator Theory on a Commutation Relation (59K, Latex2.09) Feb 22, 06
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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.

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