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Perturbation classes of semi-Fredholm operators. (English) Zbl 0541.47010
Let X and Y be Banach spaces. A bounded linear operator $$T:X\to Y$$ is called strictly singular $$(T\in {\mathfrak S}(X,Y))$$ if there is no infinite- dimensional subspace of X such that the restriction of T to X is an isomorphism. A densely defined closed linear operator S:$$X\to Y$$ is a $$\Phi_+$$-operator if the range of S is closed and its kernel is finite- dimensional. A bounded linear operator $$T:X\to Y$$ is called an admissible $$\Phi_+$$-perturbation $$(T\in {\mathfrak F}_+(X,Y))$$ if $$S+T:X\to Y$$ is a $$\Phi_+$$-operator for all $$\Phi_+$$-operators $$S:X\to Y.$$ T. Kato [J. Analyse Math. 6, 261-322 (1958; Zbl 0090.090)] proved $$\quad {\mathfrak S}(X,Y)\subset {\mathfrak F}_+(X,Y).$$ A Banach space is called weakly compactly generated if the span of some weakly compact subset of X is dense in X. The author proves the following two theorems:
Theorem A. If X is weakly compactly generated, then $${\mathfrak F}_+(X)={\mathfrak S}(X).$$
Theorem B. Let Y be weakly compactly generated. Then $${\mathfrak F}_+(X,Y)={\mathfrak S}(X,Y)$$ for all Banach spaces X if and only if Y contains at least two infinite dimensional closed subspaces M and N such that $$M\cap N=\{0\}$$ and $$M+N$$ is closed in Y.
The author also states and proves the analogous results for strictly cosingular operators and $$\Phi_-$$-perturbations.
A bounded linear operator T:$$X\to X$$ is said to be in $${\mathfrak F}(X)$$ if $$S+T$$ is a Fredholm operator on X for each Fredholm operator S on X. The author gives some examples of spaces close to $$L_ p$$-spaces for which $${\mathfrak F}(X)\neq {\mathfrak S}(X).$$
Let (G,m) be a locally compact group with its Haar measure and ($$\Gamma$$,n) be its dual group. The author proves that for $$1<p<2$$ the Fourier transform $${\mathcal F}:L_{p'}(G,m)\to L_{p'}(\Gamma,n)$$ is strictly singular and strictly cosingular where $$1/p+1/p'=1.$$
Reviewer: R.Mennicken

##### MSC:
 47A55 Perturbation theory of linear operators 47A53 (Semi-) Fredholm operators; index theories 43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
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