A ratio-dependent Leslie system with feedback controls is studied. By using a comparison theorem and constructing a suitable Lyapunov function, some sufficient conditions for the existence of a unique almost periodic solution (periodic solution) and the global attractivity of the solutions are obtained. Examples show that the obtained criteria are new, general, and easily verifiable.

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Let H be an infinite dimensional complex Hilbert space and let φ be a surjective linear map on B(H) with φ(I)−I ∈ K(H), where K(H) denotes the closed ideal of all compact operators on H. If φ preserves the set of upper semi-Weyl operators and the set of all normal eigenvalues in both directions, then φ is an automorphism of the algebra B(H). Also the relation between the linear maps preserving the set of upper semi-Weyl operators and the linear maps preserving the set of left invertible operators is considered.

A Banach space operator T ∈ B( X) may be said to be "consistent in invertibility" provided that for each S ∈ B( X), T S and S T are either both or neither invertible. The induced spectrum contributes the conditions equivalent to various forms of "Weyl's theorem".

Two variants of the Weyl spectrum are discussed. We find, for example, that if one of them coincides with the Browder spectrum then Weyl's theorem holds, and conversely for isoloid operators. Weyl [21] examined the spectra of all compact perturbations of a Hermitian operator on Hilbert space and found in 1909 that their intersection consisted precisely of those points of the spectrum which were not isolated eigenvalues of finite multiplicity. This Weyl's theorem has since been extended to hyponormal and to Toeplitz operators [3], to seminormal and other operators [1,2] and to Banach spaces operators [8,13]. Variants have been discussed by Harte and Lee [5] and Rakocevic [15]. In this note we show how Weyl's theorem follows from the equality of the Browder spectrum and a variant of the Weyl spectrum.Recall that the Weyl spectrum of a bounded linear operator T on a Banach space X is the intersection of the spectra of its compact perturbations:Equivalently λ ∈ σ w (T ) iff T − λI fails to be Fredholm of index zero. The Browder spectrum is the intersection of the spectra of its commuting compact perturbations:

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