Gramsch and Lay [8] gave spectral mapping theorems for the Dunford-Taylor calculus of a closed linear operator $T$,

\[ \widetilde{\sigma}_i(f(T)) = f(\widetilde{\sigma}_i(T)), \]

$\widetilde {\sigma }_i$

$f$

$f$

$\widetilde \sigma$

This paper is concerned with a class of non-symmetric operators, that is, 𝒥-symmetric operators, in Hilbert spaces. A sufficient condition for λ ∈ C being an element of the essential spectrum of a 𝒥-symmetric operator is given in terms of the number of linearly independent solutions of a certain homogeneous equation, and a characterization for points of the essential spectrum plus the set of all eigenvalues of a 𝒥-symmetric operator is obtained in terms of the numbers of linearly independent solutions of certain inhomogeneous equations. As direct applications, the corresponding results are obtained for singular 𝒥-symmetric Hamiltonian systems and their special forms of singular Sturm-Liouville equations with complex-valued coefficients, which enable us to study the spectra of singular 𝒥-symmetric differential expressions using numerous tools available in the fundamental theory of differential equations.

We study the effect of algebraically localized impurities on striped phases in one spatial dimension. We therefore develop a functional-analytic framework that allows us to cast the perturbation problem as a regular Fredholm problem despite the presence of the essential spectrum, caused by the soft translational mode. Our results establish the selection of jumps in wavenumber and phase, depending on the location of the impurity and the average wavenumber in the system. We also show that, for select locations, the jump in the wavenumber vanishes.

Descloux and Geymonat considered a model problem in two-dimensional magnetohydrodynamics and conjectured that the essential spectrum has an explicitly given band structure. This conjecture was recently proved by Faierman, Mennicken, and Möller by reducing the problem to that for a 2×2 block operator matrix. In a subsequent paper Faierman and Mennicken investigated the essential spectrum for the problem arising from a particular type of perturbation of precisely one of the operator entries in the matrix representation cited above of the original problem considered by Descloux and Geymonat. In this paper we extend the results of that work by investigating the essential spectrum for the problem arising from particular types of perturbations of all but one of the aforementioned operators. It remains an open question whether one can perturb the exceptional operator in such a way as to leave the essential spectrum unchanged.

We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus onphenomena related to unboundedness of the Laplacians. This includes (failure of) essentialselfadjointness, absence of essential spectrum and stochastic incompleteness.

Let T and S be quasisimilar operators on a Banach space X. A well-known result of Herrero shows that each component of the essential spectrum of T meets the essential spectrum of S. Herrero used that, for an n-multicyclic operator, the components of the essential resolvent set with maximal negative index are simply connected. We give new and conceptually simpler proofs for both of Herrero's results based on the observation that on the essential resolvent set of T the section spaces of the sheaves

are complete nuclear spaces that are topologically dual to each other. Other concrete applications of this result are given.

In this paper spectral properties of non-selfadjoint Jacobi operators $J$ which are compact perturbations of the operator $J_0=S+\rho S^*$, where $\rho\in(0,1)$ and $S$ is the unilateral shift operator in $\ell^2$, are studied. In the case where $J-J_0$ is in the trace class, Friedrichs’s ideas are used to prove similarity of $J$ to the rank one perturbation $T$ of $J_0$, i.e. $T=J_0+(\cdot,p)e_1$. Moreover, it is shown that the perturbation is of ‘smooth type’, i.e. $p\in\ell^2$ and

$$ \varlimsup_{n\rightarrow\infty}|p(n)|^{1/n}\le\rho^{1/2}. $$

When $J-J_0$ is not in the trace class, the Friedrichs method does not work and the transfer matrix approach is used. Finally, the point spectrum of a special class of Jacobi matrices (introduced by Atzmon and Sodin) is investigated.

AMS 2000 Mathematics subject classification: Primary 47B36. Secondary 47B37

In this paper, we apply coupling methods to study strong ergodicity for Markov processes, and sufficient conditions are presented in terms of the expectations of coupling times. In particular, explicit criteria are obtained for one-dimensional diffusions and birth-death processes to be strongly ergodic. As a by-product, strong ergodicity implies that the essential spectra of the generators for these processes are empty.

This paper gives a complete classification of essentially commutative C*-algebras whose essential spectrum is homeomorphic to S2n−1 by their characteristic numbers. Let 1, 2 be such two C*-algebras; then they are C*-isomorphic if and only if they have the same n-th characteristic number. Furthermore, let γn() = m then is C*-isomorphic to C*(Mzl, …, Mzn) if m = 0, is C*-isomorphic C*(Tz1, …, Tzn−1, Tznm) if m ≠ 0. Some examples are given to show applications of the classfication theorem. We finally remark that the proof of the theorem depends on a construction of a complete system of representatives of Ext(S2n−1).

In this paper, we consider the (essential) spectrum of the discrete Laplacian of an infinite graph. We introduce a new quantity for an infinite graph, in terms of which we give new lower bound estimates of the (essential) spectrum and give also upper bound estimates when the infinite graph is bipartite. We give sharp estimates of the (essential) spectrum for several examples of infinite graphs.

A vector $x$ in a Banach space $\cal B$ is called hypercyclic for a bounded linear operator $T: \cal B \rightarrow \cal B $ if the orbit $\{T^n x : n \geq 1 \}$ is dense in $\cal B$. We prove that if $T$ is hereditarily hypercyclic and its essential spectrum meets the closed unit disk, then there is an infinite-dimensional closed subspace consisting, except for zero, entirely of hypercyclic vectors for $T$. The converse is also true even if $T$ is not hereditarily hypercyclic. In this way, we improve and extend to Banach spaces a recent result for Hilbert spaces. We also investigate the corresponding problem for supercyclic operators. In this case we obtain a description of the norm closure of the class of all supercyclic operators that have an infinite-dimensional closed subspace of hypercyclic vectors. Moreover, for certain kinds of supercyclic operators we can characterize when they have an infinite-dimensional closed subspace of supercyclic vectors. 1991 Mathematics Subject Classification: 47B38, 47B99.

In this paper we study the problem of $L^p$-independence of the spectrum of second-order elliptic operators with measurable coefficients. A new technique is developed to treat the problem in cases when the kernel of the corresponding semigroup does not satisfy upper bounds of Gaussian type and the semigroup itself exists only in a certain interval of the $L^p$-scale. This allows the authors to treat second-order elliptic operators of divergence type with singular coefficients in the main part and singular lower-order terms. A criterion of $L^p$-independence obtained in the paper applies also to certain classes of higher-order elliptic operators. The problem of stability of the essential spectrum is also studied. It is shown, under some mild conditions on the coefficients of the elliptic operators, that its essential spectrum is the same as the one of the Laplacian. 1991 Mathematics Subject Classification: 35P99, 35B25, 35J15.

Let A and B be (not necessarily bounded) linear operators on a Banach lattice E such that |(s – B)-1x|≤ (s – A)-1|x| for all x in E and sufficiently large s ∈ R. The main purpose of this paper is to investigate the relation between the spectra σ(B) and σ(A) of B and A, respectively. We apply our results to study asymptotic properties of dominated C0-semigroups.

We investigate the relationship between the peripheral spectrum of a positive operator T on a Banach lattice E and the peripheral spectrum of the operators S dominated by T, that is, ]Sx] ≤ T]x] for all x ε E. This can be applied to obtain inheritance results for asymptotic properties of dominated operators.