Distortion of Banach spaces and supermultiplicative operational quantities (Q1046519)

This paper studies supermultiplicativity properties of certain operator quantities that characterize the class of upper semi-Fredholm operators on Banach spaces. Let \(X\) and \(Y\) be Banach spaces and \(T: X \to Y\) a bounded linear operator. Define \(\text{in}(T) = \inf_{M} \| T|_M\|\), \(\text{isj}(T) = \inf_{M \subset E} \sup_{N \subset M} j(T|_N)\) and \(s^*j(T) = \sup_{P \subset E} j(T|_P)\). Here, \(M\) and \(N\) range over closed infinite-dimensional subspaces of \(E\), \(P\) ranges over closed subspaces having finite codimension in \(E\), and \(j(U) = \inf \{ \| Ux\|: \| x\| = 1\}\) is the injection modulus of an operator \(U\). The author shows that the quantities \(\text{isj}(\cdot)\) and \(s^*j(\cdot)\) are supermultiplicative, while \(\text{in}(\cdot)\) does not have this property (even allowing a uniform constant). The latter result depends on the example due to \textit{Th.\,Schlumprecht} [Isr.\ J.\ Math.\ 76, No.\,1--2, 81--95 (1991; Zbl 0796.46007)] of an arbitrarily distortable Banach space. The author also discusses the related operator quantities \(\alpha (T)\) and \(\beta (T)\), where \(\alpha(T)\) (respectively, \(\beta(T)\)) is the greatest of the constants \(c \geq 0\) such that \(c \cdot \text{in}(S) \leq \text{in}(TS)\) (respectively, \(c\cdot \text{in}(S) \leq \text{in}(ST)\)) holds for all compatible operators \(S\) satisfying \(\text{in}(S) > 0\).