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arXiv · 1811.09659

Relatively bounded operators and the operator E-norms (addition to arXiv:1806.05668)

M. E. Shirokov
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Abstract

In this brief note we describe relations between the well known notion of a relatively bounded operator and the operator E-norms considered in [arXiv:1806.05668]. We show that the set of all $\sqrt{G}$-bounded operators equipped with the E-norm induced by a positive operator $G$ is the Banach space of all operators with finite E-norm and that the $\sqrt{G}$-bound is a continuous seminorm on this space. We also show that the set of all $\sqrt{G}$-infinitesimal operators (operators with zero $\sqrt{G}$-bound) equipped with the E-norm induced by a positive operator $G$ is the completion of the algebra $B(H)$ of bounded operators w.r.t. this norm. Some properties of $\sqrt{G}$-infinitesimal operators are considered.

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M. E. Shirokov. 2018-11-23. Relatively bounded operators and the operator E-norms (addition to arXiv:1806.05668). https://arxiv.org/abs/1811.09659

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