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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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