Smooth noncompact operators from \(C(K)\), \(K\) scattered
The following result is due to Pełczyński: If \(X\) is a Banach space, \(K\) is a scattered compact and \(P:C(K)\rightarrow X\) is a noncompact polynomial operator, then \(c_{0}\hookrightarrow X\). The paper under review investigates Pełczyński type results for general \(C^{1,u}\)-smooth operators. The motivation of the paper is the following question that was suggested by previous works of the second author [\textit{P.\,Hajék}, Isr.\ J.\ Math.\ 104, 17--27 (1998; Zbl 0940.46023); ibid.\ 107, 237--252 (1998; Zbl 0921.46020)]: Question 1: Let \(X\) be a Banach space, \(K\) be a scattered compact and \(T:B_{C(K)}\rightarrow X\) be a \(C^{1,u}\)-smooth noncompact operator. In this case, is it true that \(c_{0}\hookrightarrow X?\) The authors investigate the following variants of the previous question: Question 2: Let \(X\) be a Banach space, \(K\) be a scattered compact and \(T:B_{C(K)}\rightarrow X\) be a \(C^{1,u}\)-smooth noncompact operator. Does there exist \(c_{0}\simeq Y\hookrightarrow C(K)\) such that \(T|_{Y}\) is noncompact? Question 3: Let \(X\) be a Banach space, \(K\) be a scattered compact and \(T:B_{C(K)}\rightarrow X\) be a \(C^{1,u}\)-smooth noncompact operator. Does there exist a sequence \(\{u_{n}\}_{n=1}^{\infty}\) in \(B_{C(K)}\) such that both \(\{u_{n}\}_{n=1}^{\infty}\) and \(\{T(u_{n})\}_{n=1}^{\infty}\) are equivalent to the canonical basis of \(c_{0}?\) Solutions to Questions 1, 2 and 3 are investigated and discussed in the present paper. In particular, the following results are proved: - If \(X\) is a Banach space, \(K\) is a scattered compact and \(T:B_{C(K)}\rightarrow X\) is a noncompact \(C^{1,u}\)-smooth operator, then there exists an affine subspace \(c_{0}\simeq Y\subset C(K)\) such that \(T|_{Y\cap B_{C(K)}}\) is noncompact. - If \(X\) is a Banach space, \(K\) is a scattered compact and \(P:C(K)\rightarrow X\) is a noncompact polynomial operator, then there exists a sequence \(\{v_{n}\}_{n=0}^{\infty}\) in \(B_{C(K)}\) such that both \(\{v_{n}-v_{0} \}_{n=1}^{\infty}\) and \(\{P(v_{n})-P(v_{0})\}_{n=1}^{\infty}\) are equivalent to the canonical basis of \(c_{0}.\) The paper also presents interesting and relevant examples of smooth operators. For example, the authors give a striking example of a \(C^{1,1}\)-smooth noncompact operator \(T\) from \(B_{c_{0}}\) to a certain \(X\hookrightarrow l_{\infty}\) with the property that there is no sequence \(\{y^{n}\}_{n=0}^{\infty}\) in \(T(B_{c_{0}})\) such that \(\{y^{n}-y^{0}\}_{n=1}^{\infty }\) is equivalent to the canonical basis of \(c_{0}\).
scientific article; zbMATH DE number 5279132