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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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