On operator bands

Studia Mathematica (2000)

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

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Abstract

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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 $p\left(A,B\right)={\left(AB-BA\right)}^{2}$ has a special role in these considerations.

How to cite

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Drnovšek, 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

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1. [1] J. B. Conway, A Course in Functional Analysis, Springer, 1990. Zbl0706.46003
2. [2] R. Drnovšek, An irreducible semigroup of idempotents, Studia Math. 125 (1997), 97-99. Zbl0886.47005
3. [3] P. Fillmore, G. W. MacDonald, M. Radjabalipour and H. Radjavi, Principal-ideal bands, Semigroup Forum 59 (1999), 362-373. Zbl0938.20044
4. [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. [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. [6] L. Livshits, G. W. MacDonald, B. Mathes and H. Radjavi, On band algebras, ibid., to appear.
7. [7] M. Petrich, Lectures in Semigroups, Akademie-Verlag, Berlin, and Wiley, London, 1977.

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