Operators that attain their minima (Q744425)

In [Integral Equations Oper. Theory 72, No. 2, 179--195 (2012; Zbl 1257.47022)], the authors of the present paper studied Hilbert space operators which achieve their operator norms on the unit sphere. In the paper under review, they concentrate on the class of bounded linear operators on a complex Hilbert space or on a subspace of it, which attain their minima on the unit sphere. An operator \(T \in B(H,K)\) is said to satisfy the property \(\mathcal{N}^*\) if there exists an element \(x\) in the unit sphere of \(H\) such that \([T]=\inf\{\|Ty\|: \|y\|=1\}= \|Tx\|\). Any compact operator with \([T]>0\) satisfies \(\mathcal{N}^*\). In fact, the injective property plays an important role in this investigation. It is said that \(T\) satisfies the \(\mathcal{AN}^*\) property if, for all closed subspaces \(\{0\} \neq M \subseteq H\), \(T|_M\) satisfies the property \(\mathcal{N}^*\). The authors study several nice properties of operators satisfying \(\mathcal{N}^*\) or \(\mathcal{AN}^*\). In particular, it is shown that, although any projection satisfies the property \(\mathcal{N}^*\), it does not necessarily satisfy the \(\mathcal{AN}^*\) property.