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 $S_e$....