## Operational quantities derived from the minimum modulus.(English)Zbl 1119.47014

Let $$X$$, $$Y$$ be infinite dimensional Banach spaces. For two operational quantities $$a,b\:L(X,Y)\to\mathbb R^+$$ we write $$a\leq b$$ iff $$a(T)\leq b(T)$$ for each $$T\in L(X,Y)$$. Given an operational quantity $$a$$ and denoting by $$J_M$$ the canonical inclusion of a subspace $$M$$ into $$X$$, the following new quantities can be derived: $s^*a(T):=\sup\{\,a(TJ_P)\mid P\;\text{finite codimensional subspace of $$X$$}\},$
$sa(T):=\sup\{\,a(TJ_M)\mid M\;\text{infinite dimensional subspace of $$X$$}\},$
$i^*a(T):=\inf\{\,a(TJ_P)\mid P\;\text{finite codimensional subspace of $$X$$}\},$
$ia(T):=\inf\{\,a(TJ_M)\mid M\;\text{infinite dimensional subspace of $$X$$}\}.$
The authors prove some properties of these quantities, derived from the minimum modulus $$\gamma(T):=\inf_{x\notin N(T)}\| Tx\| /\operatorname{dist}(x,N(T))$$ ($$N(T)$$ is the null space of $$T$$). Particularly, they show that:
1) $$i\gamma(T)=j(T):=\inf\{\,\| Tx\| \mid x\in X,\| x\| =1\,\}$$;
2) $$sj(T)\leq s\gamma(T)\leq2sj(T)$$ if $$N(T)$$ is finite dimensional; and
3) $$s\gamma(T)=\| T\|$$ if $$N(T)$$ is infinite dimensional.
Two of the corollaries are that $$T$$ is strictly singular iff $$si\gamma(T)=0$$, and that $$T$$ is an upper semi-Fredholm operator iff $$is\gamma(T)>0$$.

### MSC:

 47A53 (Semi-) Fredholm operators; index theories
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