Timeline for Compact operators on $\ell^1$

8 events

when toggle format	what		by	license	comment
Jul 14, 2017 at 21:11	vote	accept	BaoLing
Jul 14, 2017 at 21:07	comment	added	Yemon Choi		Regarding your edit, I think the problem is that the spectral containment you mention doesn't really say anything about the multiplicity of points in the spectrum. So the points which are non-zero eigenvalues for your compact operator on $\ell^2$ might still be eigenvalues for the operator on $\ell^1$, but with "infinite multiplicity" now.
Jul 14, 2017 at 16:33	comment	added	Bill Johnson		To get an example it is enough to observe that you can find vectors $x$ in $\ell_1$ s.t. the ratio of $\\|x\\|_1 \cdot \\|x\\|_\infty $ to $\\|x\\|_2^2$ is arbitrarily large. Consider, for example, $x_n :=n^{1/2} e_1 + \sum_{k=2}^n e_k$; the desired ratio is about $n^{1/2}$. Take $\sum_{n\in S} n^{-3/2} x_n\otimes x_n$ where the $S$ is infinite, $\sum_{n\in S} n^{-1/2} < \infty$, and the sum is a direct sum.
Jul 14, 2017 at 14:47	history	edited	BaoLing	CC BY-SA 3.0	added 665 characters in body
Jul 14, 2017 at 12:53	answer	added	Matthew Daws		timeline score: 5
Jul 14, 2017 at 8:05	history	edited	BaoLing	CC BY-SA 3.0	added 199 characters in body
Jul 13, 2017 at 20:37	history	edited	BaoLing	CC BY-SA 3.0	added 41 characters in body
Jul 13, 2017 at 20:25	history	asked	BaoLing	CC BY-SA 3.0