Abstract. The left Drazin spectrum and the Drazin spectrum coincide with the upper semi-B-Browder spectrum and the B-Browder spectrum, respectively. We also prove that some spectra coincide whenever T or T * satisfies the single-valued extension property.
SVEP Polaroid operatorThe property (w) is a variant of Weyl's theorem, for a bounded operator T acting on a Banach space. In this note we consider the preservation of property (w) under a finite rank perturbation commuting with T , whenever T is polaroid, or T has analytical coreThe preservation of property (w) is also studied under commuting nilpotent or under injective quasi-nilpotent perturbations. The theory is exemplified in the case of some special classes of operators.In this paper we continue the study of the class of linear bounded operators defined on Banach spaces that verify property (w), a variant of Weyl's theorem introduced by V. Rakočević in [23] and studied in a more recent paper [8]. The preservation of property (w) under certain classes of perturbations has been investigated in [3,4,7]. In this paper we give further results on the preservation of property (w) in some special cases and improve previous results. Moreover, the theory is applied to several classes of operators. We begin by given some preliminary definitions and basic results.Let X be an infinite-dimensional complex Banach space and denote by L( X) the algebra of all bounded linear operators on X . A bounded operator T ∈ L( X) is said to be an upper semi-Fredholm operators if α(T ) := dim ker T < ∞ and T (X) is closed, while T ∈ L( X) is said to be lower semi-Fredholm if β(T ) := codim T (X) < ∞. Let Φ + (X) and Φ − (X) denote the class of all upper semi-Fredholm operators. The index of a semi-Fredholm operator is defined as ind T := α(T ) − β(T ). T ∈ L( X) is said to be a Fredholm operator if T ∈ Φ + (X) ∩ Φ − (X). The upper semi-Weyl operators W + (X) are defined as the class of upper semi-Fredholm operators having ind T 0. The lower semi-Weyl operators W − (X) are defined as the class of lower semi-Fredholm operators having ind T 0. The class of Weyl operators is defined byThese classes of operators generate the following spectra: the Weyl spectrum defined by σ w (T ) := λ ∈ C: λI − T / ∈ W (X) , the upper semi-Weyl spectrum (in literature called also Weyl essential approximate point spectrum) defined by σ uw (T ) := λ ∈ C: λI − T / ∈ W + (X) ,
A bounded linear operator T ∈ L(X) on a Banach space X is said to satisfy "Browder's theorem" if the Browder spectrum coincides with the Weyl spectrum. T ∈ L(X) is said to satisfy "a-Browder's theorem" if the upper semi-Browder spectrum coincides with the approximate point Weyl spectrum. In this note we give several characterizations of operators satisfying these theorems. Most of these characterizations are obtained by using a localized version of the single-valued extension property of T . In the last part we shall give some characterizations of operators for which "Weyl's theorem" holds. (2000). Primary 47A10 47A11; Secondary 47A53 47A55. Mathematics Subject Classification
Seasonal changes of the two dominant Zooplankton species in Lake Trasimeno, Bosmina longirostris and Daphnia galeata, was studied to evaluate factors regulating plankton production and the trophic state of the lake. Weekly samples were taken from January to December 1989. Parameters of population structure (total density, and densities of eggs, juveniles, and adults) and changes in body size were studied in relation to some environmental factors. Multiple regression models were used to analyze density and body size variations to evaluate the effects of temperature and food on growth of the two populations. The statistical models showed a significant influence of temperature and food on the seasonal patterns of density and body size of both species. In comparison with previous years, in 1989 a sharp decrease in the average body size of D. galeata was observed; this was determined primarily by the scarcity of food, rather than by predation. On the other hand, B. longirostris increased in size, being favoured by its feeding flexibility and by the composition of phytoplankton, which was made up mainly of smallsized algae; Bosmina thus became numerically dominant over Daphnia.
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