The irreducibility in ordered Banach algebras

The irreducibility in ordered Banach algebras Let A be an ordered Banach algebra. Put $$\mathbf{OI}(A)=\{b\in A: 0 \le b\le e, b^2 = b\},$$ where e is a unit of A. An element z ≥ 0 is said to be order continuous if $${b_\alpha\downarrow 0}$$ implies $${b_\alpha z \downarrow 0}$$ and $${zb_\alpha\downarrow 0}$$ for any $${b_\alpha \in \mathbf{OI}(A)}$$ . It is shown that if E is a Dedekind complete Banach lattice then the set of all order continuous elements in L(E) coincides with the set of all positive order continuous operators on E. An algebra A is said to have a (strongly) disjunctive product if for any order continuous x and y in A(x, y ≥ 0) with xy = 0 there exists $${b \in \mathbf{OI}(A)}$$ such that xb = (e − b)y = 0. We show that the algebra L(E) has the strongly disjunctive product iff E has order continuous norm. An element $${z\in A}$$ is said to be irreducible if for every $${b \in \mathbf{OI}(A)}$$ the relation (e − b)zb = 0 implies either b = 0 or b = e. We investigate spectral properties of irreducible elements in algebras with a disjunctive product. The spectral radius r(z) is called an f-pole of the resolvent R(·, z) if 0 ≤ x ≤ z implies r(x) ≤ r(z) and if r(x) = r(z) then r(z) is a pole of R(·, x). We show that under some natural assumptions on the Banach lattice E, if $${0\le T \in L(E)}$$ then r(T) is an f-pole of R(·,T) iff r(T) is a finite-rank pole of R(·, T). We also present a theorem about the Frobenius normal form of z when r(z) is an f-pole of R(·, z). Some applications to the spectral theory of irreducible operators and the general spectral theory of positive elements are provided. In particular, we show that under some conditions 0 ≤ x < z implies r(x) < r(z). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

The irreducibility in ordered Banach algebras

Positivity , Volume 16 (1) – Mar 16, 2011

Loading next page...
 
/lp/springer_journal/the-irreducibility-in-ordered-banach-algebras-VLwyxMnsdR
Publisher
Springer Journals
Copyright
Copyright © 2011 by Springer Basel AG
Subject
Mathematics; Potential Theory; Operator Theory; Fourier Analysis; Econometrics; Calculus of Variations and Optimal Control; Optimization
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-011-0117-9
Publisher site
See Article on Publisher Site

Abstract

Let A be an ordered Banach algebra. Put $$\mathbf{OI}(A)=\{b\in A: 0 \le b\le e, b^2 = b\},$$ where e is a unit of A. An element z ≥ 0 is said to be order continuous if $${b_\alpha\downarrow 0}$$ implies $${b_\alpha z \downarrow 0}$$ and $${zb_\alpha\downarrow 0}$$ for any $${b_\alpha \in \mathbf{OI}(A)}$$ . It is shown that if E is a Dedekind complete Banach lattice then the set of all order continuous elements in L(E) coincides with the set of all positive order continuous operators on E. An algebra A is said to have a (strongly) disjunctive product if for any order continuous x and y in A(x, y ≥ 0) with xy = 0 there exists $${b \in \mathbf{OI}(A)}$$ such that xb = (e − b)y = 0. We show that the algebra L(E) has the strongly disjunctive product iff E has order continuous norm. An element $${z\in A}$$ is said to be irreducible if for every $${b \in \mathbf{OI}(A)}$$ the relation (e − b)zb = 0 implies either b = 0 or b = e. We investigate spectral properties of irreducible elements in algebras with a disjunctive product. The spectral radius r(z) is called an f-pole of the resolvent R(·, z) if 0 ≤ x ≤ z implies r(x) ≤ r(z) and if r(x) = r(z) then r(z) is a pole of R(·, x). We show that under some natural assumptions on the Banach lattice E, if $${0\le T \in L(E)}$$ then r(T) is an f-pole of R(·,T) iff r(T) is a finite-rank pole of R(·, T). We also present a theorem about the Frobenius normal form of z when r(z) is an f-pole of R(·, z). Some applications to the spectral theory of irreducible operators and the general spectral theory of positive elements are provided. In particular, we show that under some conditions 0 ≤ x < z implies r(x) < r(z).

Journal

PositivitySpringer Journals

Published: Mar 16, 2011

References

  • On order convergence of nets
    Abramovich, Y.A.; Sirotkin, G.

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off