## On locally linearly dependent operators and derivations

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- by Matej Brešar and Peter Šemrl PDF
- Trans. Amer. Math. Soc.
**351**(1999), 1257-1275 Request permission

## Abstract:

The first section of the paper deals with linear operators $T_i:U\longrightarrow V$, $i = 1,\ldots ,n$, where $U$ and $V$ are vector spaces over an infinite field, such that for every $u \in U$, the vectors $T_1 u,\ldots ,T_n u$ are linearly dependent modulo a fixed finite dimensional subspace of $V$. In the second section, outer derivations of dense algebras of linear operators are discussed. The results of the first two sections of the paper are applied in the last section, where commuting pairs of continuous derivations $d,g$ of a Banach algebra $\mathcal {A}$ such that $(dg)(x)$ is quasi–nilpotent for every $x \in \mathcal {A}$ are characterized.## References

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## Additional Information

**Matej Brešar**- Affiliation: Department of Mathematics, University of Maribor PF, Koroška 160 2000 Maribor, Slovenia
- Email: bresar@uni-mb.sl
**Peter Šemrl**- Affiliation: Department of Mathematics, University of Maribor SF, Smetanova 17 2000 Maribor, Slovenia
- Email: peter.semrl@uni-mb.sl
- Received by editor(s): February 12, 1997
- Additional Notes: The authors were supported in part by the Ministry of Science of Slovenia.
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**351**(1999), 1257-1275 - MSC (1991): Primary 15A04, 16W25, 47B47; Secondary 46H05, 47B48
- DOI: https://doi.org/10.1090/S0002-9947-99-02370-3
- MathSciNet review: 1621729