On ideals of compact operators satisfying the \(M(r,s)\)-inequality
A subspace \(X\neq\{0\}\) of a Banach space \(Y\) is said to be an ideal in \(Y\) if there exists a norm one projection \(p\) on \(Y^*\) with \(\ker p= X^\perp= \{y^*\in Y^*\mid y^*(x)= 0,\forall x\in X\}\). If, moreover, \[ \| y^*\|\geq r\| py^*\|+ s\| y^*- py^*\|,\quad \forall y^*\in Y^*, \] for given \(r,s\in(0, 1]\), we say that \(X\) is an ideal satisfying the \(M(r,s)\)-inequality in \(Y\). A net of compact operators \((k_\alpha)\) on a Banach space \(X\) is called a shrinking compact approximation of the identity if \(\lim_\alpha k_\alpha x=x\) and \(\lim_\alpha k^*_\alpha x^*= x^*\) for every \(x\in X\) and \(x^*\in X^*\). Let \({\mathcal L}\) be a subspace of \({\mathcal L}(X)\) containing \({\mathcal K}(X)\) and identity \(I\). The authors prove: Theorem: Let \(X\) be a Banach space and \(r,s\in (0,1]\) be such that \(r+s/2> 1\). Then \({\mathcal K}(X)\) is an ideal satisfying the \(M(r,s)\)-inequality in \({\mathcal L}\) if and only if \(X\) admits a shrinking compact approximation of the identity \((k_\alpha)\) such that \(\| k_\alpha\|\leq 1\) for all \(\alpha\) and \[ \varlimsup_\alpha\;\| rSk_\alpha+ sT(I- k_\alpha)\|\leq 1, \] whenever \(S,T\in B_{\mathcal L}\). (\(B_{\mathcal L}\) is the unit ball of \({\mathcal L}\)).
shrinking compact approximation of the identity
scientific article; zbMATH DE number 1215727
scientific article