# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Davis, W., Dean, D., Lin, B.: Bibasic sequences and norming basic sequences. Trans. Amer. Math. Soc.176, 89-102 (1973) · Zbl 0249.46010 · doi:10.1090/S0002-9947-1973-0313763-9  Dierolf, P., Dierolf, S.: Strict Singularity and weak compactness of the Fourier Transform. Bull. Soc. Roy. Sci. Liegè48, 438-443 (1979) · Zbl 0431.43003  Diestel, J.: Geometry of Banach Spaces-Selected Topics. Lecture Notes in Mathematics485 Berlin-Heidelberg-New York: Springer 1975 · Zbl 0307.46009  Figiel, T., Johnson, W.B., Tzafriri, L.: On Banach lattices and spaces having local unconditional structure with applications to Lorentz function, spaces. J. Approximation Theory13, 395-412 (1975) · Zbl 0307.46007 · doi:10.1016/0021-9045(75)90023-4  Gohberg, I., Markus, A., Feldman, I.: Normally solvable operators and ideals associated with them. Izv. Moldar. Fil. Akad. Nauk SSSR10, 51-69 (1960) [Russian]. Engl Transl. Amer. Math Soc. Transl.61, 63-84 (1967)  Goldberg, S.: Unbounded Linear Operators. New York-London-Sydney: McGraw-Hill 1966 · Zbl 0148.12501  Goldberg, S., Kruse, A.: The existence of compact, linear maps between Banach spaces. Proc. Amer. Math. Soc.13, 808-811 (1962) · Zbl 0113.10102 · doi:10.1090/S0002-9939-1962-0141971-7  Gurarii, V., Kadec, I.: Minimal systems and quasicomplements in Banach spaces. Dokl. Acad. Nauk SSSR145, 256-258 (1962) [Russian]. Engl. Transl.: Soviet. Math. Dokl3, 966-968 (1962)  Hagler, J., Johnson, W.B.: On Banach spaces where dual balls are not weak * sequentially compact. Israel J. Math.28, 325-330 (1977) · Zbl 0365.46019 · doi:10.1007/BF02760638  Hunt, R.: OnL(p, q)-spaces. Enseignement Math.12, 249-274 (1966)  John, K., Zizler, V.: Projections in dual weakly compactly generated Banach spaces. Studia Math.49, 41-50 (1973) · Zbl 0247.46029  Johnson, W.B.: On quasi-complements. Pacific J. Math.48, 113-118 (1973) · Zbl 0283.46008  Kadec, M.I., Pelczynski, A.: Basis, lacunary sequences and complemented subspaces in the spacesL p Studia Math.21, 161-176 (1962)  Kato, T.: Perturbation theory for nullity, deficiency and other quantities of linear operators. J. Analyse Math.6, 261-322 (1958) · Zbl 0090.09003 · doi:10.1007/BF02790238  Lindenstrauß, J., Tzafriri, L.: Classical Banach spaces I. Sequence Spaces. Berlin-Heidelberg-New York: Springer 1977 · Zbl 0362.46013  Lindenstrauß, J., Tzafriri, L.: Classical Banach spaces II. Function Spaces. Berlin-Heidelberg-New York: Springer 1979 · Zbl 0403.46022  Marti, J.: Introduction to the Theory of Bases. Berlin-Heidelberg-New York: Springer 1969 · Zbl 0191.41301  Milman, V.D.: Some properties of strictly singular operators. Functional Anal. Appl.3, 77-78 (1969) · Zbl 0179.17801 · doi:10.1007/BF01078280  Pelczynski, A.: Strictly singular and strictly cosingular operators. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.13, 31-41 (1965) · Zbl 0138.38604  Pelczynski, A.: On Banach spaces containingL 1(?) Studia Math.30, 231-246 (1968)  Pietsch, A.: Operator Ideals. Berlin: Deutscher Verlag der Wissenschaften 1978 · Zbl 0399.47039  Przeworska-Rolewicz, D., Rolewicz, S.: Equations in linear spaces. Warszawa: Polish Scientific Publishers 1968 · Zbl 0181.40501  Rosenthal, H.: On totally incomparable Banach spaces. J. Functional Analysis4, 167-175 (1969) · Zbl 0184.15004 · doi:10.1016/0022-1236(69)90010-X  Rosenthal, H.: On relatively disjoint families of measures with some applications to Banach space theory. Studia Math.37, 13-36 (1970) · Zbl 0227.46027  Rudin, W.: Fourier Analysis on Groups. New York-London: Wiley 1960 · Zbl 0099.32201  Schaefer, H.H.: Banach Lattices and Positive Operators. Berlin-Heidelberg-New York: Springer 1974 · Zbl 0296.47023  Schechter, M.: Riesz operators and Fredholm perturbations. Bull. Amer. Math. Soc.74, 1139-1144 (1968) · Zbl 0167.13301 · doi:10.1090/S0002-9904-1968-12083-X  Schechter, M.: Quantities related to strictly singular operators. Indiana Univ. Math. J.21, 1061-1071 (1972) · Zbl 0274.47007 · doi:10.1512/iumj.1972.21.21085  Singer, I.: On pseudo-complemented subspaces of Banach spaces. J. Functional Analysis13, 223-232 (1973) · Zbl 0271.46012 · doi:10.1016/0022-1236(73)90032-3  Valdivia, M.: On a class of Banach spaces. Studia Math.60, 11-13 (1977) · Zbl 0354.46012  Vladimirskii, J.I.: Strictly cosingular operators. Dokl. Akad. Nauk SSSR174, 1251-1252 (1967) [Russian]. Engl. Transl.: Soviet. Math. Dokl.8, 739-740 (1967)  Weis, L.: On perturbations of Fredholm operators inL p (?)-spaces. Proc. Amer. Math. Soc.67, 287-292 (1977) · Zbl 0377.46016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.