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 $\inf \Vert T{|}_M\Vert$ over infinite-dimensional subspaces $M$. One always has $\inn(T)\le d_+(T)$, and González and Martinón 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 $d_+\leλ\,\inn$. 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 $d_+/\inn\geΛ/2$. Consequently, on the subprojective but not uniformly subprojective space $({⨁}_n{L}_{p_n}{)}_{{\ell }_2}$ 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 $\mathrm{diag}(A,B)$ 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 $\inn(T)\leτ(T)$, so every automorphism satisfies $\inn(T)\,\inn(T^{-1})\le1$ and the same mechanism cannot work for endomorphisms.

arXiv commentExpanded revision: quantitative lifting obstruction, complete characterization on L1-spaces with equality of optimal constants, and quotient obstructions on H.I. domains, including an exact radius-one construction on the Gowers-Maurey space. Proofs, references, and exposition revised. The general converse remains open