# The upper Browder spectrum property

The upper Browder spectrum property In this note we continue our study on the upper Browder spectrum initiated in Benjamin and Mouton (Quaest. Math. 39(5), 2016). Recall that, for an element a of an ordered Banach algebra A and w.r.t. a Banach algebra homomorphism $$T: A \rightarrow B,$$ T : A → B , we have inclusions \begin{aligned} \sigma (Ta) \subseteq \beta _T(a) \subseteq \beta _T^+(a) \subseteq \sigma (a), \end{aligned} σ ( T a ) ⊆ β T ( a ) ⊆ β T + ( a ) ⊆ σ ( a ) , where $$\sigma (Ta),$$ σ ( T a ) , $$\beta _T(a),$$ β T ( a ) , $$\beta _T^+(a)$$ β T + ( a ) and $$\sigma (a)$$ σ ( a ) denote the Fredholm, Browder, upper Browder and (usual) spectra of a,  respectively (Benjamin and Mouton in Quaest. Math. 39(5), 2016). This paper concerns the following natural question: given that the spectral radius of a positive element is not in the Fredholm spectrum of the element, when will it be outside the upper Browder spectrum of that element? http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# The upper Browder spectrum property

, Volume 21 (2) – May 12, 2016
18 pages

/lp/springer_journal/the-upper-browder-spectrum-property-d808Zh0WNd
Publisher
Springer International Publishing
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-016-0405-5
Publisher site
See Article on Publisher Site

### Abstract

In this note we continue our study on the upper Browder spectrum initiated in Benjamin and Mouton (Quaest. Math. 39(5), 2016). Recall that, for an element a of an ordered Banach algebra A and w.r.t. a Banach algebra homomorphism $$T: A \rightarrow B,$$ T : A → B , we have inclusions \begin{aligned} \sigma (Ta) \subseteq \beta _T(a) \subseteq \beta _T^+(a) \subseteq \sigma (a), \end{aligned} σ ( T a ) ⊆ β T ( a ) ⊆ β T + ( a ) ⊆ σ ( a ) , where $$\sigma (Ta),$$ σ ( T a ) , $$\beta _T(a),$$ β T ( a ) , $$\beta _T^+(a)$$ β T + ( a ) and $$\sigma (a)$$ σ ( a ) denote the Fredholm, Browder, upper Browder and (usual) spectra of a,  respectively (Benjamin and Mouton in Quaest. Math. 39(5), 2016). This paper concerns the following natural question: given that the spectral radius of a positive element is not in the Fredholm spectrum of the element, when will it be outside the upper Browder spectrum of that element?

### Journal

PositivitySpringer Journals

Published: May 12, 2016

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