The upper semi-Fredholm radius, extensions of restrictions, and subprojectivity

Abstract

For a bounded operator $T$ between Banach spaces, let $d_+(T)$ be its distance to the operators that are not upper semi-Fredholm, and let $\inn(T)$ be Schechter's quantity $\inf\|T|_M\|$ over infinite-dimensional subspaces $M$. One always has $\inn(T)\le d_+(T)$, and Gonz\'alez and Martin\'on showed that the two quantities are not equivalent in general. We prove that $d_+(T)$ is the infimum of the norms of operators on the whole space that extend a restriction of $T$ to some infinite-dimensional subspace. The gap between $\inn$ and $d_+$ thus becomes an extension problem, and projection constants enter naturally. If the domain is uniformly subprojective with constant $\lambda$, then $d_+\le\lambda\,\inn$. In the other direction we prove a quantitative obstruction: a complemented Hilbertian subspace all of whose infinite-dimensional subspaces have projection constant at least $\Lambda$ produces operators with $d_+/\inn\ge\Lambda/2$. Consequently, on the subprojective but not uniformly subprojective space $(\bigoplus_nL_{p_n})_{\ell_2}$ with $p_n\to\infty$, there is no uniform comparison between $\inn$ and $d_+$. We conjecture that uniform comparison is equivalent to uniform subprojectivity. Finally, we compute $\inn$ for inverses and for shears in terms of the quantity $\tau(T)=\sup_M j(T|_M)$, and we use the arbitrary distortability of Hilbert space to show that a rectangular diagonal operator $\operatorname{diag}(A,B)$ built from two isomorphisms with $\inn(A)=\inn(B)=1$ can have arbitrarily small $\inn$. In contrast, every operator $T$ on a complex Banach space satisfies $\inn(T)\le\tau(T)$, so every automorphism satisfies $\inn(T)\,\inn(T^{-1})\le1$ and the same mechanism cannot work for endomorphisms.