On the factorization of bounded linear operators and its applications

Shamim Ansari, Purdue University

Abstract

For a compact Hausdorff space $\Omega,$ order disjointness of two operators is defined in the Riesz space L(C(Omega))$ if $ Omega$ is extremally disconnected. We define essential disjointness of two operators in $L(C(Omega)),$ for an arbitrary compact Hausdorff space $ Omega.$ We prove that if $ Omega$ contains no isolated points, then any operator {\it T\/} in $L(C(Omega))$ factoring through $c sb0$ is essentially disjoint from the identity operator. Consequently, any operator $T in L(C(Omega))$ factoring through $c sb0$ satisfies the Daugavet equation. This generalizes a result of J. Holub \lbrack 20\rbrack, which states that any operator $T in L(C (0,1))$ factoring through $c sb0$ satisfies the Daugavet equation. Recently, Y. A. Abramovich, C. D. Aliprantis, and O. Burkinshaw (3) presented a Banach lattice proof of the fact that if $\Omega$ is extremally disconnected compact Hausdorff space with no isolated points and $T\in L(C(\Omega))$ is weakly compact, then T is disjoint from the identity operator. Extending this result we prove that if $\Omega$ is any compact Hausdorff space with no isolated points and $T\in L(C(\Omega))$ is weakly compact, then T is essentially disjoint from the identity. This implies that if $S,T\in L(C(\Omega)),$ S is weakly compact, and T factors through $c\sb0$ then $S+T$ satisfies the Daugavet equation. In Chapter 3 we give a necessary and sufficient condition on a Banach space B under which the set $\Phi\sb{B}(X,Y)$ of all bounded linear operators factoring through B is a vector space for all Banach spaces X and Y. In Chapter 4 our aim is to characterize Banach spaces Y for which all bounded linear operators from a $C(\Omega)$-space into Y are compact. By making use of some known results we show that for a scattered compact Hausdorff space $\Omega,$ all bounded linear operators from $C(\Omega)$ into Y are compact if and only if Y does not contain a copy of $c\sb0.$ For a nonscattered compact Hausdorff space $\Omega,$ we prove that if all bounded linear operators from $C(\Omega)$ into Y are compact, then all bounded linear operators from $l\sp2$ into Y are compact. Next, using some known results we prove that if Y has an unconditional basis, then our necessary condition is also sufficient. Consequently, if $\Omega$ is a nonscattered compact Hausdorff space, and Y is a Banach space with an unconditional basis, then all bounded linear operators from $C(\Omega)$ into Y are compact if and only if all bounded linear operators from $l\sp2$ into Y are compact.

Degree

Ph.D.

Advisors

Aliprantis, Purdue University.

Subject Area

Mathematics

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