### A boundary value problem with a discontinuous coefficient and containing a spectral parameter in the boundary condition.

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We characterize some S-essential spectra of a closed linear relation in terms of certain linear relations of semi-Fredholm type.

In this article the essential spectrum of closed, densely defined linear operators is characterized on a large class of spaces, which possess the Dunford-Pettis property or which isomorphic to one of the spaces ${L}_{p}\left(\mathrm{\Omega}\right)$$p>1$. A practical...

We characterize the bounded linear operators T in Hilbert space which satisfy T = βI + (1-β)S where β ∈ (0,1) and S is a contraction. The characterizations include a quadratic form inequality, and a domination condition of the discrete semigroup ${\left(T\u207f\right)}_{n=1,2,...}$ by the continuous semigroup ${\left({e}^{-t(I-T)}\right)}_{t\ge 0}$. Moreover, we give a stronger quadratic form inequality which ensures that $supn\parallel T\u207f-{T}^{n+1}\parallel :n=1,2,...<\infty $. The results apply to large classes of Markov operators on countable spaces or on locally compact groups.

We give a survey on spectra for various classes of nonlinear operators, with a particular emphasis on a comparison of their advantages and drawbacks. Here the most useful spectra are the asymptotic spectrum by M. Furi, M. Martelli and A. Vignoli (1978), the global spectrum by W. Feng (1997), and the local spectrum (called “phantom”) by P. Santucci and M. Väth (2000). In the last part we discuss these spectra for homogeneous operators (of any degree), and derive a discreteness result and a nonlinear...

Let $T$ be a Banach space operator. In this paper we characterize $a$-Browder’s theorem for $T$ by the localized single valued extension property. Also, we characterize $a$-Weyl’s theorem under the condition ${E}^{a}\left(T\right)={\pi}^{a}\left(T\right),$ where ${E}^{a}\left(T\right)$ is the set of all eigenvalues of $T$ which are isolated in the approximate point spectrum and ${\pi}^{a}\left(T\right)$ is the set of all left poles of $T.$ Some applications are also given.