# Some properties of essential spectra of a positive operator, II

Some properties of essential spectra of a positive operator, II Let T be a positive operator on a Banach lattice E. Some properties of Weyl essential spectrum σ ew(T), in particular, the equality $$\sigma_{\rm ew}(T) = \bigcap\limits_{0 \le K \in \mathcal {K} (E)} \sigma(T + K)$$ , where $$\mathcal {K}(E)$$ is the set of all compact operators on E, are established. If r(T) does not belong to Fredholm essential spectrum σ ef(T), then $$r(T) \notin \sigma(T + a|T_{-1}|)$$ for every a ≠ 0, where T −1 is a residue of the resolvent R(., T) at r(T). The new conditions for which $$r(T) \notin \sigma_{\rm ef}(T)$$ implies $$r(T) \notin \sigma_{\rm ew}^-(T) = \bigcap\limits_{0 \le K \in {\mathcal K}(E) \le T} \sigma(T - K)$$ , are derived. The question when the relation $$\sigma_{\rm ew}(T) \subseteq \sigma_{\rm el}(T)$$ holds, where $$\sigma_{\rm el}(T) = \bigcap\limits_{0 \le Q \le T \atop Q \le K \in {\mathcal K}(E)}\sigma(T - Q)$$ is Lozanovsky’s essential spectrum, will be considered. Lozanovsky’s order essential spectrum is introduced. A number of auxiliary results are proved. Among them the following generalization of Nikol’sky’s theorem: if T is an operator of index zero, then T = R + K, where R is invertible, K ≥ 0 is of finite rank. Under the natural assumptions (one of them is $$r(T) \notin \sigma_{\rm ef}(T)$$ ) a theorem about the Frobenius normal form is proved: there exist T-invariant bands $$E = B_n \supseteq B_{n - 1} \supseteq \cdots \supseteq B_0 = \{0\}$$ such that if $$r(P_{D_i}TP_{D_i}) = r(T)$$ , where $$D_i = B_i \cap B_{i - 1}^{\rm d}$$ , then an operator $$P_{D_i}TP_{D_i}$$ on D i is band irreducible. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Some properties of essential spectra of a positive operator, II

, Volume 13 (1) – Nov 24, 2008
18 pages

/lp/springer_journal/some-properties-of-essential-spectra-of-a-positive-operator-ii-7MvuAqRE0U
Publisher
Springer Journals
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/s11117-008-2221-z
Publisher site
See Article on Publisher Site

### Abstract

Let T be a positive operator on a Banach lattice E. Some properties of Weyl essential spectrum σ ew(T), in particular, the equality $$\sigma_{\rm ew}(T) = \bigcap\limits_{0 \le K \in \mathcal {K} (E)} \sigma(T + K)$$ , where $$\mathcal {K}(E)$$ is the set of all compact operators on E, are established. If r(T) does not belong to Fredholm essential spectrum σ ef(T), then $$r(T) \notin \sigma(T + a|T_{-1}|)$$ for every a ≠ 0, where T −1 is a residue of the resolvent R(., T) at r(T). The new conditions for which $$r(T) \notin \sigma_{\rm ef}(T)$$ implies $$r(T) \notin \sigma_{\rm ew}^-(T) = \bigcap\limits_{0 \le K \in {\mathcal K}(E) \le T} \sigma(T - K)$$ , are derived. The question when the relation $$\sigma_{\rm ew}(T) \subseteq \sigma_{\rm el}(T)$$ holds, where $$\sigma_{\rm el}(T) = \bigcap\limits_{0 \le Q \le T \atop Q \le K \in {\mathcal K}(E)}\sigma(T - Q)$$ is Lozanovsky’s essential spectrum, will be considered. Lozanovsky’s order essential spectrum is introduced. A number of auxiliary results are proved. Among them the following generalization of Nikol’sky’s theorem: if T is an operator of index zero, then T = R + K, where R is invertible, K ≥ 0 is of finite rank. Under the natural assumptions (one of them is $$r(T) \notin \sigma_{\rm ef}(T)$$ ) a theorem about the Frobenius normal form is proved: there exist T-invariant bands $$E = B_n \supseteq B_{n - 1} \supseteq \cdots \supseteq B_0 = \{0\}$$ such that if $$r(P_{D_i}TP_{D_i}) = r(T)$$ , where $$D_i = B_i \cap B_{i - 1}^{\rm d}$$ , then an operator $$P_{D_i}TP_{D_i}$$ on D i is band irreducible.

### Journal

PositivitySpringer Journals

Published: Nov 24, 2008

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