上半Fredholm半径、限制的延拓与次投影性
The upper semi-Fredholm radius, extensions of restrictions, and subprojectivity
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中文总结 AI 辅助
该文研究Banach空间算子的上半Fredholm半径与Schechter量之间的差距，将其转化为延拓问题，证明一致次投影性下二者可比，并给出非一致情形下的定量障碍，最后计算逆与剪切的Schechter量。
中文摘要 AI 辅助
对于Banach空间之间的有界算子$T$，设$d_+(T)$为其到非上半Fredholm算子的距离，并设$\inn(T)$为Schechter量$\inf\\|T|_M\\|$，其中下确界取遍无限维子空间$M$。恒有$\inn(T)\le d_+(T)$，且González和Martinón表明这两个量在一般情况下不等价。我们证明$d_+(T)$是延拓$T$到某个无限维子空间的限制的全空间上算子范数的下确界。因此，$\inn$与$d_+$之间的差距成为一个延拓问题，投影常数自然出现。若定义域是常数为$\lambda$的一致次投影空间，则$d_+\le\lambda\\,\inn$。在另一个方向上，我们证明了一个定量障碍：一个补余Hilbert子空间，其所有无限维子空间的投影常数至少为$\Lambda$，则产生满足$d_+/\inn\ge\Lambda/2$的算子。因此，在次投影但非一致次投影空间$(\bigoplus_nL_{p_n})_{\ell_2}$（其中$p_n\to\infty$）上，$\inn$与$d_+$之间不存在一致比较。我们猜想一致比较等价于一致次投影性。最后，我们利用量$\tau(T)=\sup_M j(T|_M)$计算逆算子和剪切算子的$\inn$，并利用Hilbert空间的任意可扭曲性表明，由两个满足$\inn(A)=\inn(B)=1$的同构构造的矩形对角算子$\operatorname{diag}(A,B)$可以具有任意小的$\inn$。相反，复Banach空间上的每个算子$T$都满足$\inn(T)\le\tau(T)$，因此每个自同构都满足$\inn(T)\\,\inn(T^{-1})\le1$，且同样的机制不能用于自同态。
英文摘要
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 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 $d_+\leλ\,\inn$. In the other direction we prove a quantitative obstruction: a complemented Hilbertian subspace all of whose infinite-dimensional subspaces have projection constant at least $Λ$ produces operators with $d_+/\inn\geΛ/2$. 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 $τ(T)=\sup_M j(T|_M)$, 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 $\inn(T)\leτ(T)$, so every automorphism satisfies $\inn(T)\,\inn(T^{-1})\le1$ and the same mechanism cannot work for endomorphisms.
发表机构
- University Salhi Ahmed（萨拉希艾哈迈德大学）
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