Relatively bounded operators and the operator E-norms (addition to arXiv:1806.05668)
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.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2018
- DOI:
- 10.48550/arXiv.1811.09659
- arXiv:
- arXiv:1811.09659
- Bibcode:
- 2018arXiv181109659S
- Keywords:
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- Mathematics - Functional Analysis;
- Mathematical Physics;
- Mathematics - Operator Algebras;
- Quantum Physics
- E-Print:
- 6 pages, any comments are welcome