Timeline for Bounded operator on a normed space with empty spectrum
Current License: CC BY-SA 3.0
10 events
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| Dec 1, 2014 at 12:48 | comment | added | M.González | A version of the result mentioned by Alexander Shamov can be found in Proposition 1.B.1 (page 44) of B. Beauzamy. Introduction to Operator Theory and Invariant Subspaces. North-Holland 1988. | |
| Dec 1, 2014 at 8:00 | comment | added | Jochen Wengenroth | The general fact Alexander is referring to is sometimes called "abstract Mittag-Leffler theorem" (a version of this is e.g. in Bourbaki). The first explicit appearance however is in an article of R. Arens from 1958. | |
| Nov 28, 2014 at 17:18 | vote | accept | M.González | ||
| Nov 28, 2014 at 16:12 | history | edited | André Henriques | CC BY-SA 3.0 |
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| Nov 28, 2014 at 15:39 | comment | added | Geoff Robinson | @AlexanderShamov : Thanks for the clarification. | |
| Nov 28, 2014 at 15:17 | history | edited | André Henriques | CC BY-SA 3.0 |
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| Nov 28, 2014 at 14:09 | comment | added | Alexander Shamov | @GeoffRobinson: It's actually a general fact that if we have a sequence of spaces $B_0 \supset B_1 \supset B_2 \supset \dots$ that are complete, respectively, w.r.t. norms $\Vert \cdot \Vert_0 \le \Vert \cdot \Vert_1 \le \Vert \cdot \Vert_2 \le \dots$ and all $B_{k+1}$ are dense in $(B_k, \Vert \cdot \Vert_k)$ then $\bigcap_k B_k$ is dense in $B_0$. In this case take the norms $\Vert T^{-k} (\cdot) \Vert$ on $\mathop{\mathrm{im}} T^k$. | |
| Nov 28, 2014 at 14:00 | comment | added | Alexander Shamov | @GeoffRobinson: $\bigcap_n \mathop{\mathrm{im}} T^n$ is dense iff $\mathop{\mathrm{im}} T$ is dense iff $\ker T^\ast = 0$. | |
| Nov 28, 2014 at 13:52 | comment | added | Geoff Robinson | This looks good, but is it obvious that that subspace really is dense? | |
| Nov 28, 2014 at 13:22 | history | answered | André Henriques | CC BY-SA 3.0 |