# On operator bands

Roman Drnovšek; Leo Livshits; Gordon MacDonald; Ben Mathes; Heydar Radjavi; Peter Šemrl

Studia Mathematica (2000)

- Volume: 139, Issue: 1, page 91-100
- ISSN: 0039-3223

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topDrnovšek, Roman, et al. "On operator bands." Studia Mathematica 139.1 (2000): 91-100. <http://eudml.org/doc/216713>.

@article{Drnovšek2000,

abstract = {A multiplicative semigroup of idempotent operators is called an operator band. We prove that for each K>1 there exists an irreducible operator band on the Hilbert space $l^2$ which is norm-bounded by K. This implies that there exists an irreducible operator band on a Banach space such that each member has operator norm equal to 1. Given a positive integer r, we introduce a notion of weak r-transitivity of a set of bounded operators on a Banach space. We construct an operator band on $l^2$ that is weakly r-transitive and is not weakly (r+1)-transitive. We also study operator bands S satisfying a polynomial identity p(A, B) = 0 for all non-zero A,B ∈ S, where p is a given polynomial in two non-commuting variables. It turns out that the polynomial $p(A, B) = (A B - B A)^2$ has a special role in these considerations.},

author = {Drnovšek, Roman, Livshits, Leo, MacDonald, Gordon, Mathes, Ben, Radjavi, Heydar, Šemrl, Peter},

journal = {Studia Mathematica},

keywords = {invariant subspaces; idempotents; operator semigroups; irreducible operator band; Hilbert space; Banach space},

language = {eng},

number = {1},

pages = {91-100},

title = {On operator bands},

url = {http://eudml.org/doc/216713},

volume = {139},

year = {2000},

}

TY - JOUR

AU - Drnovšek, Roman

AU - Livshits, Leo

AU - MacDonald, Gordon

AU - Mathes, Ben

AU - Radjavi, Heydar

AU - Šemrl, Peter

TI - On operator bands

JO - Studia Mathematica

PY - 2000

VL - 139

IS - 1

SP - 91

EP - 100

AB - A multiplicative semigroup of idempotent operators is called an operator band. We prove that for each K>1 there exists an irreducible operator band on the Hilbert space $l^2$ which is norm-bounded by K. This implies that there exists an irreducible operator band on a Banach space such that each member has operator norm equal to 1. Given a positive integer r, we introduce a notion of weak r-transitivity of a set of bounded operators on a Banach space. We construct an operator band on $l^2$ that is weakly r-transitive and is not weakly (r+1)-transitive. We also study operator bands S satisfying a polynomial identity p(A, B) = 0 for all non-zero A,B ∈ S, where p is a given polynomial in two non-commuting variables. It turns out that the polynomial $p(A, B) = (A B - B A)^2$ has a special role in these considerations.

LA - eng

KW - invariant subspaces; idempotents; operator semigroups; irreducible operator band; Hilbert space; Banach space

UR - http://eudml.org/doc/216713

ER -

## References

top- [1] J. B. Conway, A Course in Functional Analysis, Springer, 1990. Zbl0706.46003
- [2] R. Drnovšek, An irreducible semigroup of idempotents, Studia Math. 125 (1997), 97-99. Zbl0886.47005
- [3] P. Fillmore, G. W. MacDonald, M. Radjabalipour and H. Radjavi, Principal-ideal bands, Semigroup Forum 59 (1999), 362-373. Zbl0938.20044
- [4] J. A. Green and D. Rees, On semigroups in which ${x}^{r}=x$, Proc. Cambridge Philos. Soc. 48 (1952), 35-40. Zbl0046.01903
- [5] L. Livshits, G. W. MacDonald, B. Mathes and H. Radjavi, Reducible semigroups of idempotent operators, J. Operator Theory 40 (1998), 35-69. Zbl0995.47002
- [6] L. Livshits, G. W. MacDonald, B. Mathes and H. Radjavi, On band algebras, ibid., to appear.
- [7] M. Petrich, Lectures in Semigroups, Akademie-Verlag, Berlin, and Wiley, London, 1977.

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