A note on the common spectral properties for bounded linear operators

Hassane Zguitti

arXiv:1903.11153·math.FA·April 2, 2019

A note on the common spectral properties for bounded linear operators

Hassane Zguitti

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TL;DR

This paper investigates spectral properties of bounded linear operators on Banach spaces, establishing conditions under which their spectra, excluding zero, are equal for certain operator products.

Contribution

It introduces new spectral equivalence results for operator products under specific algebraic conditions, expanding understanding of spectral behavior in Banach space operators.

Findings

01

Spectral equality for operator products under given conditions

02

Extension of spectral theory for regularities in Banach spaces

03

Conditions ensuring spectral invariance excluding zero

Abstract

Let $X$ and $Y$ be Banach spaces, $A : X \to Y$ and $B, C : Y \to X$ be bounded linear operators. We prove that if $A (B A)^{2} = A B A C A = A C A B A = (A C)^{2} A,$ then $σ_{*} (A C) ∖ {0} = σ_{*} (B A) ∖ {0}$ where $σ_{*}$ runs over a large of spectra originated by regularities.

Equations132

σ_{*} (A C) ∖ {0} = σ_{*} (B A) ∖ {0}

σ_{*} (A C) ∖ {0} = σ_{*} (B A) ∖ {0}

σ (A B) ∖ {0} = σ (B A) ∖ {0}

σ (A B) ∖ {0} = σ (B A) ∖ {0}

A (B A)^{2} = A B A C A = A C A B A = (A C)^{2} A .

A (B A)^{2} = A B A C A = A C A B A = (A C)^{2} A .

σ_{*} (A C) ∖ {0} = σ_{*} (B A) ∖ {0}

σ_{*} (A C) ∖ {0} = σ_{*} (B A) ∖ {0}

c_{n} (T) = dim X / (R (T) + N (T^{n})) \mbox an d c_{n}^{'} (T) = dim N (T) \cap R (T^{n}) .

c_{n} (T) = dim X / (R (T) + N (T^{n})) \mbox an d c_{n}^{'} (T) = dim N (T) \cap R (T^{n}) .

N^{\infty} (T) = n = 1 ⋃ \infty N (T^{n}) \mbox an d R^{\infty} (T) = n = 1 ⋂ \infty R (T^{n}) .

N^{\infty} (T) = n = 1 ⋃ \infty N (T^{n}) \mbox an d R^{\infty} (T) = n = 1 ⋂ \infty R (T^{n}) .

\begin{array}[]{lcl}k_{n}(T)&=&\dim((\mathcal{R}(T^{n})\cap\mathcal{N}(T))/(\mathcal{R}(T^{n+1})\cap\mathcal{N}(T)))\\ &=&\dim(\mathcal{R}(T)+\mathcal{R}(T^{n+1}))/(\mathcal{R}(T)+\mathcal{N}(T^{n})).\end{array}

\begin{array}[]{lcl}k_{n}(T)&=&\dim((\mathcal{R}(T^{n})\cap\mathcal{N}(T))/(\mathcal{R}(T^{n+1})\cap\mathcal{N}(T)))\\ &=&\dim(\mathcal{R}(T)+\mathcal{R}(T^{n+1}))/(\mathcal{R}(T)+\mathcal{N}(T^{n})).\end{array}

k (T) = n = 0 \sum \infty k_{n} (T) .

k (T) = n = 0 \sum \infty k_{n} (T) .

c (T) = n = 0 \sum \infty c_{n} (T) \mbox an d c^{'} (T) = n = 0 \sum \infty c_{n}^{'} (T) .

c (T) = n = 0 \sum \infty c_{n} (T) \mbox an d c^{'} (T) = n = 0 \sum \infty c_{n}^{'} (T) .

d i s (T) = in f {n \in Z_{+} : k_{m} (T) = 0 \mbox f or a l l m \geq n},

d i s (T) = in f {n \in Z_{+} : k_{m} (T) = 0 \mbox f or a l l m \geq n},

d i s_{e} (T) = in f {n \in Z_{+} : k_{m} (T) < \infty \mbox f or a l l m \geq n} .

d i s_{e} (T) = in f {n \in Z_{+} : k_{m} (T) < \infty \mbox f or a l l m \geq n} .

σ_{R} (T) = {λ \in C : T - λ I \in / R}

σ_{R} (T) = {λ \in C : T - λ I \in / R}

\begin{array}[]{lcl}R_{1}&=&\{T\in\mathcal{L}(X)\,:\,c(T)=0\},\\ R_{2}&=&\{T\in\mathcal{L}(X)\,:\,c(T)<\infty\},\\ R_{3}&=&\{T\in\mathcal{L}(X)\,:\mbox{ there exists }d\in\mathbb{Z}_{+}\mbox{ such that }c_{d}(T)=0\mbox{ and }\mathcal{R}(T^{d+1})\mbox{ is closed}\},\\ R_{4}&=&\{T\in\mathcal{L}(X)\,:\,c_{n}(T)<\infty,\forall n\in\mathbb{Z}_{+}\},\\ R_{5}&=&\{T\in\mathcal{L}(X)\,:\,\mbox{ there exists }d\in\mathbb{Z}_{+}\mbox{ such that }c_{d}(T)<\infty\mbox{ and }\mathcal{R}(T^{d+1})\mbox{ is closed}\},\\ R_{6}&=&\{T\in\mathcal{L}(X)\,:\,c^{\prime}(T)=0\mbox{ and }\mathcal{R}(T)\mbox{ is closed}\},\\ R_{7}&=&\{T\in\mathcal{L}(X)\,:\,c^{\prime}(T)<\infty\mbox{ and }\mathcal{R}(T)\mbox{ is closed}\},\\ R_{8}&=&\{T\in\mathcal{L}(X)\,:\,\mbox{ there exists }d\in\mathbb{Z}_{+}\mbox{ such that }c_{d}^{\prime}(T)=0\mbox{ and }\mathcal{R}(T^{d+1})\mbox{ is closed}\},\\ R_{9}&=&\{T\in\mathcal{L}(X)\,:\,c_{n}^{\prime}(T)<\infty\mbox{ for every }n\in\mathbb{Z}_{+}\mbox{ and }\mathcal{R}(T)\mbox{ is closed}\},\\ R_{10}&=&\{T\in\mathcal{L}(X)\,:\,\mbox{ there exists }d\in\mathbb{Z}_{+}\mbox{ such that }c_{d}^{\prime}(T)<\infty\mbox{ and }\mathcal{R}(T^{d+1})\mbox{ is closed}\},\\ R_{11}&=&\{T\in\mathcal{L}(X)\,:\,k(T)=0\mbox{ and }\mathcal{R}(T)\mbox{ is closed}\},\\ R_{12}&=&\{T\in\mathcal{L}(X)\,:\,k(T)<\infty\mbox{ and }\mathcal{R}(T)\mbox{ is closed}\},\\ R_{13}&=&\{T\in\mathcal{L}(X)\,:\,\mbox{ there exists }d\in\mathbb{Z}_{+}\mbox{ such that }k_{n}(T)=0\mbox{ for every }n\geq d\mbox{ and }\mathcal{R}(T^{d+1})\mbox{ is closed}\},\\ R_{14}&=&\{T\in\mathcal{L}(X)\,:\,k_{n}(T)<\infty\mbox{ for every }n\in\mathbb{Z}_{+}\mbox{ and }\mathcal{R}(T)\mbox{ is closed}\},\\ R_{15}&=&\{T\in\mathcal{L}(X)\,:\,\mbox{ there exists }d\in\mathbb{Z}_{+}\mbox{ such that }k_{n}(T)<\infty\mbox{ for every }n\geq d\mbox{ and }\mathcal{R}(T^{d+1})\mbox{ is closed}\},\\ R_{16}&=&\{T\in\mathcal{L}(X)\,:\mbox{ there exists }d\in\mathbb{Z}_{+}\mbox{ such that }c_{d}(T)=0\mbox{ and }\mathcal{R}(T)+N(T^{d})\mbox{ is closed}\},\\ R_{17}&=&\{T\in\mathcal{L}(X)\,:\mbox{ there exists }d\in\mathbb{Z}_{+}\mbox{ such that }c_{d}(T)<\infty\mbox{ and }\mathcal{R}(T)+N(T^{d})\mbox{ is closed}\},\\ R_{18}&=&\{T\in\mathcal{L}(X)\,:\,\exists d\in\mathbb{Z}_{+}\mbox{ such that }k_{n}(T)=0\mbox{ for every }n\geq d\mbox{ and }\mathcal{R}(T)+N(T^{d})\mbox{ is closed}\},\\ R_{19}&=&\{T\in\mathcal{L}(X)\,:\,\exists d\in\mathbb{Z}_{+}\mbox{ such that }k_{n}(T)<\infty\mbox{ for every }n\geq d\mbox{ and }\mathcal{R}(T)+N(T^{d})\mbox{ is closed}\}.\end{array}

\begin{array}[]{lcl}R_{1}&=&\{T\in\mathcal{L}(X)\,:\,c(T)=0\},\\ R_{2}&=&\{T\in\mathcal{L}(X)\,:\,c(T)<\infty\},\\ R_{3}&=&\{T\in\mathcal{L}(X)\,:\mbox{ there exists }d\in\mathbb{Z}_{+}\mbox{ such that }c_{d}(T)=0\mbox{ and }\mathcal{R}(T^{d+1})\mbox{ is closed}\},\\ R_{4}&=&\{T\in\mathcal{L}(X)\,:\,c_{n}(T)<\infty,\forall n\in\mathbb{Z}_{+}\},\\ R_{5}&=&\{T\in\mathcal{L}(X)\,:\,\mbox{ there exists }d\in\mathbb{Z}_{+}\mbox{ such that }c_{d}(T)<\infty\mbox{ and }\mathcal{R}(T^{d+1})\mbox{ is closed}\},\\ R_{6}&=&\{T\in\mathcal{L}(X)\,:\,c^{\prime}(T)=0\mbox{ and }\mathcal{R}(T)\mbox{ is closed}\},\\ R_{7}&=&\{T\in\mathcal{L}(X)\,:\,c^{\prime}(T)<\infty\mbox{ and }\mathcal{R}(T)\mbox{ is closed}\},\\ R_{8}&=&\{T\in\mathcal{L}(X)\,:\,\mbox{ there exists }d\in\mathbb{Z}_{+}\mbox{ such that }c_{d}^{\prime}(T)=0\mbox{ and }\mathcal{R}(T^{d+1})\mbox{ is closed}\},\\ R_{9}&=&\{T\in\mathcal{L}(X)\,:\,c_{n}^{\prime}(T)<\infty\mbox{ for every }n\in\mathbb{Z}_{+}\mbox{ and }\mathcal{R}(T)\mbox{ is closed}\},\\ R_{10}&=&\{T\in\mathcal{L}(X)\,:\,\mbox{ there exists }d\in\mathbb{Z}_{+}\mbox{ such that }c_{d}^{\prime}(T)<\infty\mbox{ and }\mathcal{R}(T^{d+1})\mbox{ is closed}\},\\ R_{11}&=&\{T\in\mathcal{L}(X)\,:\,k(T)=0\mbox{ and }\mathcal{R}(T)\mbox{ is closed}\},\\ R_{12}&=&\{T\in\mathcal{L}(X)\,:\,k(T)<\infty\mbox{ and }\mathcal{R}(T)\mbox{ is closed}\},\\ R_{13}&=&\{T\in\mathcal{L}(X)\,:\,\mbox{ there exists }d\in\mathbb{Z}_{+}\mbox{ such that }k_{n}(T)=0\mbox{ for every }n\geq d\mbox{ and }\mathcal{R}(T^{d+1})\mbox{ is closed}\},\\ R_{14}&=&\{T\in\mathcal{L}(X)\,:\,k_{n}(T)<\infty\mbox{ for every }n\in\mathbb{Z}_{+}\mbox{ and }\mathcal{R}(T)\mbox{ is closed}\},\\ R_{15}&=&\{T\in\mathcal{L}(X)\,:\,\mbox{ there exists }d\in\mathbb{Z}_{+}\mbox{ such that }k_{n}(T)<\infty\mbox{ for every }n\geq d\mbox{ and }\mathcal{R}(T^{d+1})\mbox{ is closed}\},\\ R_{16}&=&\{T\in\mathcal{L}(X)\,:\mbox{ there exists }d\in\mathbb{Z}_{+}\mbox{ such that }c_{d}(T)=0\mbox{ and }\mathcal{R}(T)+N(T^{d})\mbox{ is closed}\},\\ R_{17}&=&\{T\in\mathcal{L}(X)\,:\mbox{ there exists }d\in\mathbb{Z}_{+}\mbox{ such that }c_{d}(T)<\infty\mbox{ and }\mathcal{R}(T)+N(T^{d})\mbox{ is closed}\},\\ R_{18}&=&\{T\in\mathcal{L}(X)\,:\,\exists d\in\mathbb{Z}_{+}\mbox{ such that }k_{n}(T)=0\mbox{ for every }n\geq d\mbox{ and }\mathcal{R}(T)+N(T^{d})\mbox{ is closed}\},\\ R_{19}&=&\{T\in\mathcal{L}(X)\,:\,\exists d\in\mathbb{Z}_{+}\mbox{ such that }k_{n}(T)<\infty\mbox{ for every }n\geq d\mbox{ and }\mathcal{R}(T)+N(T^{d})\mbox{ is closed}\}.\end{array}

R_{1} \subseteq R_{2} = R_{3} \cap R_{4} \subseteq R_{3} \cup R_{4} \subseteq R_{5} \subseteq R_{13},

R_{1} \subseteq R_{2} = R_{3} \cap R_{4} \subseteq R_{3} \cup R_{4} \subseteq R_{5} \subseteq R_{13},

R_{6} \subseteq R_{7} = R_{8} \cap R_{9} \subseteq R_{8} \cup R_{9} \subseteq R_{10} \subseteq R_{13},

R_{6} \subseteq R_{7} = R_{8} \cap R_{9} \subseteq R_{8} \cup R_{9} \subseteq R_{10} \subseteq R_{13},

R_{11} \subseteq R_{12} = R_{13} \cap R_{14} \subseteq R_{13} \cup R_{14} \subseteq R_{15} .

R_{11} \subseteq R_{12} = R_{13} \cap R_{14} \subseteq R_{13} \cup R_{14} \subseteq R_{15} .

\begin{array}[]{lcl}R_{3}&=&\{T\in\mathcal{L}(X)\,:\,dsc(T)<\infty\mbox{ and }\mathcal{R}(T^{dsc(T)+1})\mbox{ is closed}\},\\ R_{5}&=&\{T\in\mathcal{L}(X)\,:\,dsc_{e}(T)<\infty\mbox{ and }\mathcal{R}(T^{dsc_{e}(T)+1})\mbox{ is closed}\},\\ R_{8}&=&\{T\in\mathcal{L}(X)\,:\,asc(T)<\infty\mbox{ and }\mathcal{R}(T^{asc(T)+1})\mbox{ is closed}\},\\ R_{10}&=&\{T\in\mathcal{L}(X)\,:\,asc_{e}(T)<\infty\mbox{ and }\mathcal{R}(T^{asc_{e}(T)+1})\mbox{ is closed}\},\\ R_{13}&=&\{T\in\mathcal{L}(X)\,:\,dis(T)<\infty\mbox{ and }\mathcal{R}(T^{dis(T)+1})\mbox{ is closed}\},\\ R_{15}&=&\{T\in\mathcal{L}(X)\,:\,dis_{e}(T)<\infty\mbox{ and }\mathcal{R}(T^{dis_{e}(T)+1})\mbox{ is closed}\}.\end{array}

\begin{array}[]{lcl}R_{3}&=&\{T\in\mathcal{L}(X)\,:\,dsc(T)<\infty\mbox{ and }\mathcal{R}(T^{dsc(T)+1})\mbox{ is closed}\},\\ R_{5}&=&\{T\in\mathcal{L}(X)\,:\,dsc_{e}(T)<\infty\mbox{ and }\mathcal{R}(T^{dsc_{e}(T)+1})\mbox{ is closed}\},\\ R_{8}&=&\{T\in\mathcal{L}(X)\,:\,asc(T)<\infty\mbox{ and }\mathcal{R}(T^{asc(T)+1})\mbox{ is closed}\},\\ R_{10}&=&\{T\in\mathcal{L}(X)\,:\,asc_{e}(T)<\infty\mbox{ and }\mathcal{R}(T^{asc_{e}(T)+1})\mbox{ is closed}\},\\ R_{13}&=&\{T\in\mathcal{L}(X)\,:\,dis(T)<\infty\mbox{ and }\mathcal{R}(T^{dis(T)+1})\mbox{ is closed}\},\\ R_{15}&=&\{T\in\mathcal{L}(X)\,:\,dis_{e}(T)<\infty\mbox{ and }\mathcal{R}(T^{dis_{e}(T)+1})\mbox{ is closed}\}.\end{array}

σ_{R_{i}} (A C) ∖ {0} = σ_{R_{i}} (B A) ∖ {0} \mbox f or 1 \leq i \leq 19.

σ_{R_{i}} (A C) ∖ {0} = σ_{R_{i}} (B A) ∖ {0} \mbox f or 1 \leq i \leq 19.

A B A (C A - I)^{k} = (A B - I)^{k} A B A \mbox an d A C A (B A - I)^{k} = (A C - I)^{k} A C A .

A B A (C A - I)^{k} = (A B - I)^{k} A B A \mbox an d A C A (B A - I)^{k} = (A C - I)^{k} A C A .

A B A Q (C A - I) = Q (A B - I) A B A \mbox an d A C A Q (B A - I) = Q (A C - I) A C A .

A B A Q (C A - I) = Q (A B - I) A B A \mbox an d A C A Q (B A - I) = Q (A C - I) A C A .

c_{n} (A C - I) = c_{n} (B A - I) \mbox f or a l l n \in Z_{+} .

c_{n} (A C - I) = c_{n} (B A - I) \mbox f or a l l n \in Z_{+} .

Γ_{A C A} : R ((B A - I)^{n}) / R ((B A - I)^{n + 1}) \to R ((A C - I)^{n}) / R ((A C - I)^{n + 1})

Γ_{A C A} : R ((B A - I)^{n}) / R ((B A - I)^{n + 1}) \to R ((A C - I)^{n}) / R ((A C - I)^{n + 1})

Γ_{A C A} (x + R ((B A - I)^{n + 1})) = A C A x + R ((A C - I)^{n + 1}) .

Γ_{A C A} (x + R ((B A - I)^{n + 1})) = A C A x + R ((A C - I)^{n + 1}) .

(B A)^{4} x = B A B A C A C A x \in R ((B A - I)^{n + 1}) .

(B A)^{4} x = B A B A C A C A x \in R ((B A - I)^{n + 1}) .

\begin{array}[]{lcl}x&=&(BA)^{4}x-((BA)^{4}-I)x\\ &=&(BA)^{4}x-((BA)^{3}+(BA)^{2}+(BA)+I)(BA-I)x\\ &=&(BA)^{4}x-((BA)^{3}+(BA)^{2}+(BA)+I)(BA-I)^{n+1}z\\ &=&(BA)^{4}x-(BA-I)^{n+1}(((BA)^{3}+(BA)^{2}+(BA)+I)z)\in\mathcal{R}((BA-I)^{n+1}).\end{array}

\begin{array}[]{lcl}x&=&(BA)^{4}x-((BA)^{4}-I)x\\ &=&(BA)^{4}x-((BA)^{3}+(BA)^{2}+(BA)+I)(BA-I)x\\ &=&(BA)^{4}x-((BA)^{3}+(BA)^{2}+(BA)+I)(BA-I)^{n+1}z\\ &=&(BA)^{4}x-(BA-I)^{n+1}(((BA)^{3}+(BA)^{2}+(BA)+I)z)\in\mathcal{R}((BA-I)^{n+1}).\end{array}

c_{n} (B A - I) \leq c_{n} (A C - I) .

c_{n} (B A - I) \leq c_{n} (A C - I) .

c_{n} (C A - I) \leq c_{n} (A B - I) .

c_{n} (C A - I) \leq c_{n} (A B - I) .

\begin{array}[]{lcl}c_{n}(BA-I)&\leq&c_{n}(AC-I)\\ &=&c_{n}(CA-I)\mbox{ (\cite[cite]{[\@@bibref{}{ZZ0}{}{}, Lemma 3.9]}}\\ &\leq&c_{n}(AB-I)\mbox{ by (\ref{eq2c})}\\ &=&c_{n}(BA-I)\mbox{ (\cite[cite]{[\@@bibref{}{ZZ0}{}{}, Lemma 3.9]}}.\end{array}

\begin{array}[]{lcl}c_{n}(BA-I)&\leq&c_{n}(AC-I)\\ &=&c_{n}(CA-I)\mbox{ (\cite[cite]{[\@@bibref{}{ZZ0}{}{}, Lemma 3.9]}}\\ &\leq&c_{n}(AB-I)\mbox{ by (\ref{eq2c})}\\ &=&c_{n}(BA-I)\mbox{ (\cite[cite]{[\@@bibref{}{ZZ0}{}{}, Lemma 3.9]}}.\end{array}

σ_{a cs} (T) = {λ \in C : a sc (T) = \infty} \mbox an d σ_{a cs}^{e} (T) = {λ \in C : a sc (T) .

σ_{a cs} (T) = {λ \in C : a sc (T) = \infty} \mbox an d σ_{a cs}^{e} (T) = {λ \in C : a sc (T) .

σ_{*} A C ∖ {0} = σ_{*} B A ∖ {0}, \mbox f or σ_{*} \in {σ_{a sc}, σ_{a cs}^{e}} .

σ_{*} A C ∖ {0} = σ_{*} B A ∖ {0}, \mbox f or σ_{*} \in {σ_{a sc}, σ_{a cs}^{e}} .

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Full text

A note on the common spectral properties for bounded linear operators

Hassane Zguitti

Hassane Zguitti: Department of Mathematics, Dhar El Mahraz Faculty of Science, Sidi Mohamed Ben Abdellah University, BO 1796 Fes-Atlas, 30003 Fez Morocco.

[email protected]

Abstract.

Let $X$ and $Y$ be Banach spaces, $A\,:\,X\rightarrow Y$ and $B,\,C\,:\,Y\rightarrow X$ be bounded linear operators. We prove that if $A(BA)^{2}=ABACA=ACABA=(AC)^{2}A,$ then

[TABLE]

where $\sigma_{*}$ runs over a large of spectra originated by regularities.

Key words and phrases:

Jacobson’s lemma, common spectral properties, regularity

2010 Mathematics Subject Classification:

47A10, 47A53, 47A55.

1. Introduction

Throughout this paper $\mathcal{L}(X,Y)$ denotes the set of all bounded linear operators acting from a complex Banach space $X$ into another one, $Y$ , and $\mathcal{L}(X)$ is a short for $\mathcal{L}(X,X)$ . Given two operators $A\in\mathcal{L}(X,Y)$ and $B\in\mathcal{L}(Y,X)$ , Jacobson’s Lemma asserts that

[TABLE]

where $\sigma(\cdot)$ denotes the ordinary spectrum.

Several works have been devoted to equality (1.1) by showing that $AB-I$ and $BA-I$ share many spectral properties. See [2, 3, 5, 6, 13, 15, 16, 18, 19] and the references therein. Barnes in [2] extended (1.1) to other part of the spectrum and showed that $AB-I$ and $BA-I$ share some spectral properties. In [3], Benhida and Zerouali investigated equation (1.1) for various Taylor joint spectra. For $A$ and $B$ satisfying $ABA=A^{2}$ and $BAB=B^{2}$ , Schmoeger [15, 16] and Duggal [7] showed that $A$ , $B$ , $AB$ and $BA$ share spectral properties. Corach et al. [6] investigated common properties for $ac-1$ and $ba-1$ where $a,b$ and $c$ are elements in associative ring such that $aba=aca$ . For bounded linear operators $A$ , $B$ and $C$ , Zeng and Zhong [19] studied spectral properties for $AC$ and $BA$ under the condition $ABA=ACA$ . If $C=I$ in the last condition, one can retrieve Schmoeger’s result. For operators $A$ , $B$ , $C$ and $D$ satisfying $ACD=DBD$ and $BDA=ACA$ , Yan and Fang [17] investigated spectral properties for $AC$ and $BD$ . Recently, [5] studied common properties for $ac$ and $ba$ for elements in a ring satisfying $a(ba)^{2}=abaca=acaba=(ac)^{2}a$ .

The paper is a continuation of [5] and [20]. The aim of this paper is to extend recent results to bounded linear operators $A\in\mathcal{L}(X,Y)$ and $B,C\in\mathcal{L}(Y,X)$ satisfying

[TABLE]

In section two we give basic definitions and notation which we need in the sequel. Section 3 is devoted to the main results of the paper. In Theorem 3.1 we prove that if $A\in\mathcal{L}(X,Y)$ and $B,C\in\mathcal{L}(Y,X)$ satisfy $A(BA)^{2}=ABACA=ACABA=(AC)^{2}A,$ then

[TABLE]

where $\sigma_{*}$ runs over a large of spectra originated by regularities.

2. Basic definitions and notations

For an operator $T\in\mathcal{L}(X)$ , let $\mathcal{N}(T)$ and $\mathcal{R}(T)$ stand for the kernel, respectively the range of $T$ . An operator $T\in\mathcal{L}(X)$ is said to be an upper semi-Fredholm operator if ${\mathcal{R}}(T)$ is closed and $\dim\mathcal{N}(T)<\infty$ , and $T$ is said to be a lower semi-Fredholm operator if $\mbox{codim}\mathcal{N}(T)<\infty$ . One says that $T$ is a Fredholm operator if $\dim\mathcal{N}(T)<\infty$ and $\mbox{codim}\mathcal{N}(T)<\infty$ . If $T$ is either upper or lower semi-Fredholm then $T$ is said semi-Fredholm operator. In this case the index of $T$ is defined by $\mbox{ind}(T)=\dim\mathcal{N}(T)-\dim\mathcal{R}(T)$ .

The ascent of $T$ , $asc(T)$ , is the smallest nonnegative integer $n$ for which $N(T^{n})=N(T^{n+1})$ , i.e.; $asc(T)=\inf\{n\in\mathbb{Z}_{+}\,:\,\mathcal{N}(T^{n})=\mathcal{N}(T^{n+1})\}$ . If no such integer exists, we shall say that $T$ has infinite ascent. In a similar way, the descent of $T$ , $dsc(T)$ , is defined by $dsc(T)=\inf\{n\in\mathbb{Z}_{+}\,:\,\mathcal{R}(T^{n})=\mathcal{R}(T^{n+1})\}$ and if no such integer exists, we shall say that $T$ has infinite descent. We say that $T$ is left Drazin invertible if $asc(T)<\infty$ and $\mathcal{R}(T^{asc(T)+1})$ is closed and $T$ is right Drazin invertible if $dsc(T)<\infty$ and $\mathcal{R}(T^{dsc(T)})$ is closed. If $T$ is both left and right Drazin invertible, then $T$ is said to be Drazin invertible ; which is equivalent to $asc(T)=dsc(T)<\infty$ (see [1]). One says that $T$ is upper semi-Browder if $T$ is upper semi-Fredholm with finite ascent, and $T$ is lower semi-Browder if $T$ is lower semi-Fredholm with finite descent. If $T$ is both upper and lower semi-Browder then $T$ is said to be Browder operator (see [14]).

For each $n\in\mathbb{Z}_{+}$ , let $c_{n}(T)=\dim R(T^{n})/R(T^{n+1})$ and $c_{n}^{\prime}(T)=\dim N(T^{n+1})/N(T^{n}).$ It was proved in [8, Lemma 3.2] that for every $n$ , we have

[TABLE]

It is easy to see that $\{c_{n}(T)\}$ and $\{c_{n}^{\prime}(T)\}$ are decreasing sequences and $dsc(T)=\inf\{n\in\mathbb{Z}_{+}\,:\,c_{n}(T)=0\},$ $asc(T)=\inf\{n\in\mathbb{Z}_{+}\,:\,c_{n}^{\prime}(T)=0\}.$

Following [12], the essential descent $dsc_{e}(T)$ of $T$ is defined by $dsc_{e}(T)=\inf\{n\in\mathbb{Z}_{+}\,:\,c_{n}(T)<\infty\},$ and the essential ascent $asc_{e}(T)$ of $T$ is defined by $asc_{e}(T)=\inf\{n\in\mathbb{Z}_{+}\,:\,c_{n}^{\prime}(T)<\infty\},$ where the infimum over the empty set is taken to be infinite.

Let $\mathcal{N}^{\infty}(T)$ and $\mathcal{R}^{\infty}(T)$ denote the hyper-kernel and the hyper-range of $T$ defined by

[TABLE]

One says that $T$ is semi-regular if $\mathcal{R}(T)$ is closed and $\mathcal{N}^{\infty}(T)\subseteq\mathcal{R}(T)$ .

For each $n\in\mathbb{Z}_{+}$ , $T\in\mathcal{L}(X)$ induces a linear maps $\Gamma_{n}$ from the space $\mathcal{R}(T^{n})/\mathcal{R}(T^{n+1})$ into $\mathcal{R}(T^{n+1})/\mathcal{R}(T^{n+2})$ . The dimension of the null space of $\Gamma_{n}$ will be denoted by $k_{n}(T)$ , i.e., $k_{n}(T)=\dim\mathcal{N}(\Gamma_{n}).$ It follows from [8, Theorem 3.7] that for every $n$ ,

[TABLE]

Let

[TABLE]

Then it follows from [8, Theorem 3.7] that $k(T)=\dim\mathcal{N}(T)/(\mathcal{N}(T)\cap\mathcal{R}^{\infty}(T))=\dim(\mathcal{R}(T)+\mathcal{N}^{\infty}(T))/\mathcal{R}(T).$ The stable nullity $c(T)$ and the stable defect $c^{\prime}(T)$ of $T$ are defined by

[TABLE]

Then we have $c(T)=\dim X/\mathcal{R}^{\infty}(T)\mbox{ and }c^{\prime}(T)=\dim\mathcal{R}^{\infty}(T).$

According to [11], the degree of stable iteration of $T\in\mathcal{L}(X)$ is defined by

[TABLE]

and the degree of essential stable iteration of $T$ ([18]) is defined is

[TABLE]

Definition 2.1.

Let $R$ be a non-empty subset of ${\mathcal{L}}(X)$ . $R$ is called a regularity if it satisfies the following two conditions:

i)

if $n\in\mathbb{N}$ , then $A\in R$ if and only if $A^{n}\in R$ ;

ii)

if $A,B,C$ and $D$ are mutually commuting operators in $\mathcal{L}(X)$ such that $AC+BD=I$ , then $AB\in R\mbox{ if and only if }A\in R\mbox{ and }B\in R.$

A regularity $R\subset{\mathcal{L}}(X)$ assigns to each $T\in{\mathcal{L}}(X)$ a subset of $\mathbb{C}$ defined by

[TABLE]

and called the spectrum of $T$ corresponding to the regularity $R$ . We note that every regularity $R$ contains all invertible operators, so that $\sigma_{R}(T)\subseteq\sigma(T)$ . In general, $\sigma_{R}(T)$ is neither compact nor non-empty (see [10, 12, 14]).

The regularities $R_{i}$ , where $\,1\leq i\leq 15$ , were introduced and studied in [10, 12, 14] but are in a different form. Regularity $R_{18}$ was introduced by [4], while $R_{16},R_{17}$ and $R_{19}$ were introduced by [18].

Definition 2.2.

[TABLE]

We have

[TABLE]

It was proved in [18, Proposition 2.7] that

[TABLE]

The operators of $R_{1},R_{2},R_{3},R_{4}$ and $R_{5}$ are surjective, lower semi-Browder, right Drazin invertible, lower semi-Fredholm and right essentially Drazin invertible operators, respectively. The operators of $R_{6},R_{7},R_{8},R_{9}$ and $R_{10}$ are bounded below, upper semi-Browder, left Drazin invertible, upper semi-Fredholm and left essentially Drazin invertible operators, respectively. The operators of $R_{11},R_{12}$ and $R_{13}$ are semi-regular, essentially semi-regular and quasi-Fredholm operators. The operators of $R_{18}$ are the operators with eventual topological uniform descent.

3. Main results

The following is our main result.

Theorem 3.1.

Let $A\in\mathcal{L}(X,Y)$ and $B,C\in\mathcal{L}(Y,X)$ such that $A(BA)^{2}=ABACA=ACABA=(AC)^{2}A.$ Then

[TABLE]

The proof of our main result uses several auxiliary lemmas.

Lemma 3.2.

*Let $A\in\mathcal{L}(X,Y)$ and $B,C\in\mathcal{L}(Y,X)$ such that $A(BA)^{2}=ABACA=ACABA=(AC)^{2}A.$ Let $Q$ be a polynomial. Then we have

$ABA\mathcal{R}(Q(CA-I))\subseteq\mathcal{R}(Q(AB-I))$ ;
$ABA\mathcal{N}(Q(CA-I)\subseteq\mathcal{N}(Q(AB-I))$ ;
$ACA\mathcal{R}(Q(BA-I))\subseteq\mathcal{R}(Q(AC-I))$ ;
$ACA\mathcal{N}(Q(BA-I))\subseteq\mathcal{N}(Q(AC-I))$ .*

Proof.

It is easy to see that for each $k\in\mathbb{Z}_{+}$ ,

[TABLE]

Then

[TABLE]

Let $x$ belongs to $\mathcal{R}(Q(CA-I))$ . Then there exists some $y\in X$ such that $Q(CA-I)y=x$ . Hence it follows from (3.1) that $ABAx=ABAQ(CA-I)x=Q(AB-I)ABAx$ which belongs to $\mathcal{R}(Q(AB-I))$ . Thus $ABA\mathcal{R}(Q(CA-I))\subseteq\mathcal{R}(Q(AB-I))$ .
Let $x\in\mathcal{N}(Q(CA-I))$ . Then $Q(CA-I)x=0$ . It follows from (3.1) that $Q(AB-I)ABAx=ABAQ(CA-I)x=0$ . Thus $ABAx\in\mathcal{N}(Q(AB-I))$ .

Using (3.2), 3) and 4) go similarly. ∎

Lemma 3.3.

Let $A\in\mathcal{L}(X,Y)$ and $B,C\in\mathcal{L}(Y,X)$ such that $A(BA)^{2}=ABACA=ACABA=(AC)^{2}A.$ Then

[TABLE]

In particular, $c(AC-I)=c(BA-I).$

Proof.

Let

[TABLE]

be the linear application defined by

[TABLE]

Since $ACA\mathcal{R}((BA-I)^{n})\subseteq\mathcal{R}((AC-I)^{n})$ by Lemma 3.2, part 3), then $\Gamma_{ACA}$ is well defined. We shall show that $\Gamma_{ACA}$ is injective.

Let $x\in\mathcal{R}((BA-I)^{n})$ such that $\Gamma_{ACA}(x)=0$ . Then $ACAx\in\mathcal{R}((AC-I)^{n+1})$ . Hence $CACAx\in\mathcal{R}((CA-I)^{n+1})$ . From Lemma 3.2, part 1), we have $ABACACAx\in\mathcal{R}((AB-I)^{n+1}).$ Then

[TABLE]

Since $x\in\mathcal{R}((BA-I)^{n})$ then $x=(BA-I)^{n}z$ for some $z\in X$ . Hence

[TABLE]

Thus $\Gamma_{ACA}$ is injective and consequently

[TABLE]

In similar way, we show that

[TABLE]

Finally,

[TABLE]

Therefore $c_{n}(BA-I)=c_{n}(AC-I)$ for all $n\in\mathbb{Z}_{+}$ . In particular, $c(AC-I)=c(BA-I).$ ∎

For $T\in\mathcal{L}(X)$ , let $\sigma_{asc}(T)$ and $\sigma_{asc}^{e}(T)$ be, respectively, the ascent spectrum and the descent spectrum of $T$ defined by

[TABLE]

The following is an immediate consequence of Lemma 3.3.

Corollary 3.4.

Let $A\in\mathcal{L}(X,Y)$ and $B,C\in\mathcal{L}(Y,X)$ such that $A(BA)^{2}=ABACA=ACABA=(AC)^{2}A.$ Then

[TABLE]

Lemma 3.5.

Let $A\in\mathcal{L}(X,Y)$ and $B,C\in\mathcal{L}(Y,X)$ such that $A(BA)^{2}=ABACA=ACABA=(AC)^{2}A.$ Then

[TABLE]

In particular, $c^{\prime}(AC-I)=c^{\prime}(BA-I).$

Proof.

Let

[TABLE]

be the linear application defined by

[TABLE]

Since $ACA\mathcal{N}((BA-I)^{n+1})\subseteq\mathcal{N}((AC-I)^{n+1})$ by Lemma 3.2, part 4), then $\Psi_{ACA}$ is well defined.

Now we show that $\Psi_{ACA}$ is injective. Let $x\in\mathcal{N}((BA-I)^{n+1})$ such that $\Psi_{ACA}(x)=0$ , which means that $ACAx\in\mathcal{N}((AC-I)^{n})$ . Hence $CACAx\in\mathcal{N}((CA-I)^{n})$ . It follows from Lemma 3.2, part ii), that $ABACACAx\in\mathcal{N}((AB-I)^{n}).$ Then

[TABLE]

Hence

[TABLE]

Which implies that $\Psi_{ACA}$ is injective and then

[TABLE]

Similarly, we prove that

[TABLE]

Finally,

[TABLE]

Therefore $c_{n}^{\prime}(BA-I)=c_{n}^{\prime}(AC-I)$ for all $n\in\mathbb{Z}_{+}$ . In particular, $c^{\prime}(AC-I)=c^{\prime}(BA-I).$ ∎

For $T\in\mathcal{L}(X)$ let $\sigma_{dsc}(T)$ and $\sigma_{dsc}^{e}(T)$ be respectively the descent spectrum and the essential descent spectrum of $T$ defined by

[TABLE]

Then the following is an immediate consequence of Lemma 3.5

Corollary 3.6.

Let $A\in\mathcal{L}(X,Y)$ and $B,C\in\mathcal{L}(Y,X)$ such that $A(BA)^{2}=ABACA=ACABA=(AC)^{2}A.$ Then

[TABLE]

Lemma 3.7.

Let $A\in\mathcal{L}(X,Y)$ and $B,C\in\mathcal{L}(Y,X)$ such that $A(BA)^{2}=ABACA=ACABA=(AC)^{2}A.$ Then

[TABLE]

In particular, $k(AC-I)=k(BA-I).$

Proof.

Let $\Phi_{ACA}$ be the linear application from $\mathcal{R}(BA-I)+\mathcal{N}((BA-I)^{n+1})/\mathcal{R}(BA-I)+\mathcal{N}((BA-I)^{n})$ to $\mathcal{R}(AC-I)+\mathcal{N}((AC-I)^{n+1})/\mathcal{R}(AC-I)+\mathcal{N}((AC-I)^{n})$ defined by

[TABLE]

Since, by Lemme 3.2, parts 3) and 4),

[TABLE]

then $\Phi_{ACA}$ is well defined.

We prove that $\Phi_{ACA}$ is injective. Let $x\in\mathcal{R}(BA-I)+\mathcal{N}((BA-I)^{n+1})$ such that $\Phi_{ACA}(x)=0$ . Then $ACAx\in\mathcal{R}(AC-I)+\mathcal{N}((AC-I)^{n})$ . So, there exist some $y\in\mathcal{R}(BA-I)$ and $z\in\mathcal{N}((AC-I)^{n})$ such that $ACAx=y+z$ . Then $CACAx=Cy+Cz\in\mathcal{R}(CA-I)+\mathcal{N}((CA-I)^{n})$ . Thus by Lemma 3.2, parts 1) and 2), we get that $ABACACAx\in\mathcal{R}(AB-I)+\mathcal{N}((AB-I)^{n})$ and consequently $(BA)^{4}x=BABACACAx\in\mathcal{R}(BA-I)+\mathcal{N}((BA-I)^{n})$ . Thus

[TABLE]

Hence $\Phi_{ACA}$ is injective. Thus

[TABLE]

In similar way, we show that

[TABLE]

Therefore,

[TABLE]

∎

Lemma 3.8.

Let $A\in\mathcal{L}(X,Y)$ and $B,C\in\mathcal{L}(Y,X)$ such that $A(BA)^{2}=ABACA=ACABA=(AC)^{2}A.$ Then for all $n\in\mathbb{Z}_{+}$ , $\mathcal{R}((AC-I)+\mathcal{N}((AC-I)^{n})$ is closed if and only if $\mathcal{R}(BA-I)+\mathcal{N}((BA-I)^{n})$ is closed.

In particular $\mathcal{R}(AC-I)$ is closed if and only if $\mathcal{R}(BA-I)$ is closed.

Proof.

Assume that $\mathcal{R}(AC-I)+\mathcal{N}((AC-I)^{n})$ is closed. Let $\{x_{p}\}$ be a sequence in $\mathcal{R}(BA-I)+\mathcal{N}((BA-I)^{n})$ which converges to $x\in X$ . Then $ACAx_{p}$ converge to $ACAx$ . Since $ACA(\mathcal{R}(BA-I)+\mathcal{N}((BA-I)^{n}))\subset\mathcal{R}(AC-I)+\mathcal{N}((AC-I)^{n})$ by Lemma 3.2, part 3) and 4), then $ACAx_{p}$ belongs to $\mathcal{R}((AC-I)+\mathcal{N}((AC-I)^{n})$ . Since $\mathcal{R}(AC-I)+\mathcal{N}((AC-I)^{n})$ is closed and $ACAx_{p}$ converges to $ACAx$ .

[TABLE]

Thus

[TABLE]

Therefore $\mathcal{R}(BA-I))+\mathcal{N}((BA-I)^{n})$ is closed.

The opposite implication goes similarly. ∎

Lemma 3.9.

Let $A\in\mathcal{L}(X,Y)$ and $B,C\in\mathcal{L}(Y,X)$ such that $A(BA)^{2}=ABACA=ACABA=(AC)^{2}A.$ Then for all $n\in\mathbb{N}$ , $\mathcal{R}((AC-I)^{n})$ is closed if and only if $\mathcal{R}(BA-I)^{n})$ is closed.

Proof.

As in the presentation before [2, Proposition], for each $n\in\mathbb{N}$ there exists $B_{n}$ and $C_{n}\in\mathcal{L}(Y,X)$ such that

[TABLE]

Indeed, we have $B_{n}=\displaystyle{\sum_{k=1}^{n}}(-1)^{k-1}({}_{k}^{n})B(AB)^{k-1}$ and $C_{n}=\displaystyle{\sum_{k=1}^{n}}(-1)^{k-1}({}_{k}^{n})(CA)^{k-1}C$ . It is easy to check that

[TABLE]

Then it follows from Lemma 3.8 that $\mathcal{R}((AC-I)^{n})$ is closed if and only if $\mathcal{R}((BA-I)^{n})$ is closed. ∎

Proof of Theorem 3.1 : The proof follows at once from Lemmas 3.2-3.9.

4. Applications and concluding remarks

A bounded operator $T\in\mathcal{L}(X)$ is said to be upper semi-Weyl operator if $T$ is upper semi-Fredholm with $ind(T)\leq 0$ , and $T$ is said to be lower semi-Weyl operator if $T$ is lower semi-Fredholm with $ind(T)\geq 0$ . If $T$ is both upper and lower semi-Fredholm then $T$ is said to Weyl operator. Then $T$ is weyl operator precisely when $T$ is a Fredholm operator with index zero. The upper semi-Weyl spectrum $\sigma_{uw}(T)$ , the *lower semi-Weyl spectrum * $\sigma_{lw}(T)$ and the Weyl spectrum $\sigma_{w}(T)$ of $T$ are defined by

[TABLE]

From Lemma 3.3 and Lemma 3.5 we deduce the following result

Proposition 4.1.

Let $A\in\mathcal{L}(X,Y)$ and $B,C\in\mathcal{L}(Y,X)$ such that $A(BA)^{2}=ABACA=ACABA=(AC)^{2}A.$ Then

[TABLE]

An operator $T\in\mathcal{L}(X)$ is said to be Riesz operator if $T-\lambda I$ is a Fredholm operator for all $0\neq\lambda\in\mathbb{C}$ . Then the following proposition is an immediate consequence of Theorem 3.1

Proposition 4.2.

Let $A\in\mathcal{L}(X,Y)$ and $B,C\in\mathcal{L}(Y,X)$ such that $A(BA)^{2}=ABACA=ACABA=(AC)^{2}A.$ Then $AC$ is a Riesz operator if and only if $BA$ is a Riesz operator.

Following [21], an operator $T\in\mathcal{L}(X)$ is said to be generalized Drazin-Riesz operator if there exists $S\in\mathcal{L}(X)$ such that

[TABLE]

The operator $S$ is called a generalized Drazin-Riesz inverse of $T$ .

Theorem 4.3.

Let $A\in\mathcal{L}(X,Y)$ and $B,C\in\mathcal{L}(Y,X)$ such that $A(BA)^{2}=ABACA=ACABA=(AC)^{2}A.$ Then $AC$ is generalized Drazin-Riesz invertible if and only if $BA$ is generalized Drazin-Riesz invertible. In this case, if $S$ is a generalized Drazin-Riesz inverse of $AC$ then $BS^{2}A$ is a generalized Drazin-inverse of $BA$ .

Proof.

Assume that $AC$ is generalized Drazin-Riesz invertible. then there exists $S\in\mathcal{L}(X)$ such that $S(AC)=(AC)S$ , $S(AC)S=S$ and $(AC)^{2}S-AC$ is Riesz. Set $T=BS^{2}A$ and we shall show that

[TABLE]

For the first equality, we have

[TABLE]

For the second,

[TABLE]

Set $P=ACS-I=SAC-I$ . Then

[TABLE]

Hence it remains to show that $BPA$ is a Riesz operator. We have

[TABLE]

In the same way, one can prove that

[TABLE]

Since $(PA)C=(AC)^{2}S-AC$ is a Riesz operator by assumption, then it follows from Proposition 4.2 that $B(PA)$ is a Riesz operator. Therefore $BA$ is generalized Drazin-Riesz invertible and $BS^{2}A$ is a generalized Drazin-inverse of $BA$ .

In similar way, we prove the opposite implication. ∎

*Remark 4.4**.*

If $A$ and $B\in\mathcal{L}(X)$ such that $ABA=A^{2}$ and $BAB=B^{2}$ , then

[TABLE]

and

[TABLE]

Then it follows from (4.1) and (4.2 that $A$ , $B$ , $BA$ and $AB$ share above spectral properties. So we retrieve the results of [7].

In the following two examples, the common spectral properties for $AC$ and $BA$ can only followed directly from the above results, but not from the corresponding ones in [15, 16, 7, 19, 9].

*Example 4.5**.*

Let $P$ be a non trivial idempotent on $X$ . Let $A$ , $B$ and $C$ defined on $X\oplus X\oplus X$ by

[TABLE]

Then $A(BA)^{2}=ABACA=ACABA=(AC)^{2}A$ , while $ABA\neq ACA$ and $BAB\neq B^{2}$ .

*Example 4.6**.*

Let $A$ and $B$ be as in Example 4.5 and let $C$ be defined on $X\oplus X\oplus X$ by

[TABLE]

Then $A(BA)^{2}=ABACA=ACABA=(AC)^{2}A$ , while $ABA\neq ACA$ and $BAB\neq B^{2}$ .

Bibliography21

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] P. Aiena, M. T. Biondi and C. Carpintero, On Drazin invertibility , Proc. Amer. Math. Soc. 136 (8) (2008), 2839-2848.
2[2] B. A. Barnes, Common operator properties of the linear operators RS and SR . Proc. Amer. Math. Soc., 126, (1998), 1055–1061.
3[3] C. Benhida and E. H. Zerouali, On Taylor and other joint spectra for commuting n 𝑛 n -tuples of operators . J. Math. Anal. Appl., 326 (2007), 521-532.
4[4] M. Berkani, Restriction of an operator to the range of its powers , Studia Math. 140 (2000), 163-175.
5[5] H. Chen and M. Sheibani, Cline’s formula for g-Drazin inverses , Arxiv math. 1805.06133 v 1
6[6] G. Corach, B. Duggal and R. Harte, Extensions of Jacobson’s lemma . Comm. Algebra, 41 (2013), 520-531.
7[7] B. P. Duggal, Operator equations A B A = A 2 𝐴 𝐵 𝐴 superscript 𝐴 2 ABA=A^{2} and B A B = B 2 𝐵 𝐴 𝐵 superscript 𝐵 2 BAB=B^{2} , Fun. Anal. Approx. Comp. 3 (1) (2011), 9-18.
8[8] S. Grabiner, Uniform ascent and descent of bounded operators . J. Math. Soc. Japan, 34 (1982), 317-337.

TL;DR

Contribution

Findings

Abstract

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A note on the common spectral properties for bounded linear operators

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

2. Basic definitions and notations

Definition 2.1**.**

Definition 2.2**.**

3. Main results

Theorem 3.1**.**

Lemma 3.2**.**

Proof.

Lemma 3.3**.**

Proof.

Corollary 3.4**.**

Lemma 3.5**.**

Proof.

Corollary 3.6**.**

Lemma 3.7**.**

Proof.

Lemma 3.8**.**

Proof.

Lemma 3.9**.**

Proof.

4. Applications and concluding remarks

Proposition 4.1**.**

Proposition 4.2**.**

Theorem 4.3**.**

Proof.

Remark 4.4*.*

Example 4.5*.*

Example 4.6*.*

Definition 2.1.

Definition 2.2.

Theorem 3.1.

Lemma 3.2.

Lemma 3.3.

Corollary 3.4.

Lemma 3.5.

Corollary 3.6.

Lemma 3.7.

Lemma 3.8.

Lemma 3.9.

Proposition 4.1.

Proposition 4.2.

Theorem 4.3.

*Remark 4.4**.*

*Example 4.5**.*

*Example 4.6**.*