The paper is devoted to solvability conditions for linear elliptic problems with non-Fredholm operators. We show that the operator becomes normally solvable with a finite-dimensional kernel on properly chosen subspaces. In the particular case of a scalar equation we obtain necessary and sufficient solvability conditions. These results are used to apply the implicit function theorem for a nonlinear elliptic problem; we demonstrate the persistence of travelling wave solutions to spatially periodic...

Simple examples of bounded domains $D\subset {𝐑}^3$ are considered for which the presence of peculiar corners and edges in the boundary $\delta D$ causes that the double layer potential operator acting on the space $𝒮\left(\delta D\right)$ of all continuous functions on $\delta D$ can for no value of the parameter $\alpha$ be approximated (in the sub-norm) by means of operators of the form $\alpha I+T$ (where $I$ is the identity operator and $T$ is a compact linear operator) with a deviation less then $\left|\alpha \right|$; on the other hand, such approximability turns out to be possible for...

We are interested of the Newton type mixed problem for the general second order semilinear evolution equation. Applying Nikolskij’s decomposition theorem and general Fredholm operator theory results, the present paper yields sufficient conditions for generic properties, surjectivity and bifurcation sets of the given problem.

Burgos, Kaidi, Mbekhta and Oudghiri [J. Operator Theory 56 (2006)] provided an affirmative answer to a question of Kaashoek and Lay and proved that an operator F is of power finite rank if and only if ${\sigma }_{dsc}(T+F)={\sigma }_{dsc}\left(T\right)$ for every operator T commuting with F. Later, several authors extended this result to the essential descent spectrum, left Drazin spectrum and left essential Drazin spectrum. In this paper, using the theory of operators with eventual topological uniform descent and the technique used by Burgos et...

Let A stand for a Toeplitz operator with a continuous symbol on the Bergman space of the polydisk ${}^N$ or on the Segal-Bargmann space over ${ℂ}^N$. Even in the case N = 1, the spectrum Λ(A) of A is available only in a few very special situations. One approach to gaining information about this spectrum is based on replacing A by a large “finite section”, that is, by the compression $A_n$ of A to the linear span of the monomials $z_1^{k_1}...z_N^{k_N}:0\le k_j\le n$. Unfortunately, in general the spectrum of $A_n$ does not mimic the spectrum of A as...

To every elliptic SG pseudo-differential operator with positive orders, we associate the minimal and maximal operators on $L^p\left(ℝⁿ\right)$, 1 < p < ∞, and prove that they are equal. The domain of the minimal ( = maximal) operator is explicitly computed in terms of a Sobolev space. We prove that an elliptic SG pseudo-differential operator is Fredholm. The essential spectra of elliptic SG pseudo-differential operators with positive orders and bounded SG pseudo-differential operators with orders 0,0 are computed....

Let A(·) be a regular function defined on a connected metric space G whose values are mutually commuting essentially Kato operators in a Banach space. Then the spaces $R^{\infty }\left(A\left(z\right)\right)$ and $\overline{N^{\infty }\left(A\left(z\right)\right)}$ do not depend on z ∈ G. This generalizes results of B. Aupetit and J. Zemánek.

We prove the stability under compact perturbations of the algebraic index of a Fredholm linear relation with closed range acting between normed spaces. Our main tool is a result concerning the stability of the index of a complex of Banach spaces under compact perturbations.

Upper semi-Fredholm operators and tauberian operators in Banach spaces admit the following perturbative characterizations [6], [2]: An operator T: X --&gt; Y is upper semi-Fredholm (tauberian) if and only if for every compact operator K: X --&gt; Y the kernel N(T+K) is finite dimensional (reflexive). In [7] Tacon introduces an intermediate class between upper semi-Fredholm operators and tauberian operators, the supertauberian operators, and he studies this class using non-standard analysis....

Pour un opérateur T borné sur un espace de Hilbert dans lui-même, nous montrons que $\gamma \left(\pi \right(T\left)\right)=sup\gamma (T+K):Kopérateurcompact$, où γ est la conorme (the reduced minimum modulus) et π(T) est la classe de T dans l’algèbre de Calkin. Nous montrons aussi que ce supremum est atteint. D’autre part, nous montrons que les opérateurs semi-Fredholm caractérisent les points de continuité de l’application T → γ (π(T)).

We study the problem of approximation by the sets S + K(H), $S_e$, V + K(H) and $V_e$ where H is a separable complex Hilbert space, K(H) is the ideal of compact operators, $S=L\in B\left(H\right):L*L=I$ is the set of isometries, V = S ∪ S* is the set of maximal partial isometries, $S_e=L\in B\left(H\right):\pi (L*)\pi \left(L\right)=\pi \left(I\right)$ and $V_e=S_e\cup S_e*$ where π : B(H) → B(H)/K(H) denotes the canonical projection. We also prove that all the relevant distances are attained. This implies that all these classes are closed and we remark that $V_e=V+K\left(H\right)$. We also show that S + K(H) is both closed and open in $S_e$....