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)≤ 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 λ, then d+≤λ 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 Λ produces operators with d+/inn≥Λ/2. Consequently, on the subprojective but not uniformly subprojective space (⨁nLpn)ℓ2 with pn→∞, 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 τ(T)=supM j(T|M), and we use the arbitrary distortability of Hilbert space to show that a rectangular diagonal operator 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)≤τ(T), so every automorphism satisfies inn(T) inn(T−1)≤1 and the same mechanism cannot work for endomorphisms.
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BibTeX
@misc{azzouz2026upper,
title = {{The upper semi-Fredholm radius, extensions of restrictions, and subprojectivity}},
author = {Abdelhalim Azzouz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03835},
eprint = {2610.03835},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03835},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The upper semi-Fredholm radius, extensions of restrictions, and subprojectivity
AU - Abdelhalim Azzouz
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.03835
UR - https://arxiv.org/abs/2610.03835
AB - 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)≤ 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 λ, then d_+≤λ 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 Λ produces operators with d_+/inn≥Λ/2. Consequently, on the subprojective but not uniformly subprojective space (⨁_nL_p_n)_ℓ_2 with p_n→∞, 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 τ(T)=sup_M j(T|_M), and we use the arbitrary distortability of Hilbert space to show that a rectangular diagonal operator 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)≤τ(T), so every automorphism satisfies inn(T) inn(T^−1)≤1 and the same mechanism cannot work for endomorphisms.
ER -