## On locally linearly dependent operators and derivations

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- by Matej Brešar and Peter Šemrl
- Trans. Amer. Math. Soc.
**351**(1999), 1257-1275 - DOI: https://doi.org/10.1090/S0002-9947-99-02370-3
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## 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

- S. A. Amitsur,
*Generalized polynomial identities and pivotal monomials*, Trans. Amer. Math. Soc.**114**(1965), 210–226. MR**172902**, DOI 10.1090/S0002-9947-1965-0172902-9 - Bernard Aupetit,
*A primer on spectral theory*, Universitext, Springer-Verlag, New York, 1991. MR**1083349**, DOI 10.1007/978-1-4612-3048-9 - Jeffrey Bergen,
*Derivations in prime rings*, Canad. Math. Bull.**26**(1983), no. 3, 267–270. MR**703394**, DOI 10.4153/CMB-1983-042-2 - Matej Brešar,
*Derivations mapping into the socle. II*, Proc. Amer. Math. Soc.**126**(1998), no. 1, 181–187. MR**1416078**, DOI 10.1090/S0002-9939-98-03993-8 - Matej Brešar and Peter emrl,
*Derivations mapping into the socle*, Math. Proc. Cambridge Philos. Soc.**120**(1996), no. 2, 339–346. MR**1384473**, DOI 10.1017/S0305004100074892 - P. Fillmore, C. Laurie, and H. Radjavi,
*On matrix spaces with zero determinant*, Linear and Multilinear Algebra**18**(1985), no. 3, 255–266. MR**828407**, DOI 10.1080/03081088508817691 - Cahit Arf,
*Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper*, J. Reine Angew. Math.**181**(1939), 1–44 (German). MR**18**, DOI 10.1515/crll.1940.181.1 - Tadasi Nakayama,
*On Frobeniusean algebras. I*, Ann. of Math. (2)**40**(1939), 611–633. MR**16**, DOI 10.2307/1968946 - Vladimír Müller,
*Kaplansky’s theorem and Banach PI-algebras*, Pacific J. Math.**141**(1990), no. 2, 355–361. MR**1035448** - Edward C. Posner,
*Derivations in prime rings*, Proc. Amer. Math. Soc.**8**(1957), 1093–1100. MR**95863**, DOI 10.1090/S0002-9939-1957-0095863-0 - C. J. K. Batty,
*Perturbations of ground states of type $\textrm {I}$ $C^{\ast }$-algebras*, Proc. Amer. Math. Soc.**78**(1980), no. 4, 539–544. MR**556628**, DOI 10.1090/S0002-9939-1980-0556628-9 - Heydar Radjavi and Peter Rosenthal,
*Invariant subspaces*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77, Springer-Verlag, New York-Heidelberg, 1973. MR**0367682** - A. M. Sinclair,
*Continuous derivations on Banach algebras*, Proc. Amer. Math. Soc.**20**(1969), 166–170. MR**233207**, DOI 10.1090/S0002-9939-1969-0233207-X - S. Minakshi Sundaram,
*On non-linear partial differential equations of the hyperbolic type*, Proc. Indian Acad. Sci., Sect. A.**9**(1939), 495–503. MR**0000089**

## Bibliographic 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