9
$\begingroup$

Excerpt from "A Hilbert-space problem book"It was mentioned on Page 916 Problem 7 of Halmos's "Ten Problems in Hilbert space" that there is a proof for "Every compact quasinilpotent operator is the limit of nilpotent ones" by R. G. Douglas that uses the upper semi-continuity of the spectrum.

Does anyone have any references or suggestions for the full proof?

To be more specific, the part I'm having trouble with is "it follows (triangular form) that $B_n$ has the form $C_n+D_n$ where $C_n$ is nilpotent and $D_n$ is diagonal, with $\lVert D_n\rVert < \frac{1}{n} $."

I'm not sure where the triangular form came from.

Thank you for any help.

Improve this question
$\endgroup$
5
  • 1
    $\begingroup$ I think the OP refers to the paper by Halmos in Bull. Amer. Math. Soc. 76 (1970), 887-933. $\endgroup$
    – J.J. Green
    Commented Mar 9, 2021 at 9:07
  • $\begingroup$ Halmos' comment makes sense if for $B_n$ you use $P_nAP_n$, where the $P_n$ are orthogonal projections onto the span of the first $n$ basis vectors of an orthonormal basis and you work with $B_n$ on the range $E_n$ of $P_n$. $\endgroup$
    – Bill Johnson
    Commented Aug 7, 2021 at 18:31
  • 1
    $\begingroup$ Does anyone know anything about this question for Banach spaces other than Hilbert space? AFAIK, it could be true in every Banach space that is not isomorphic to a Hilbert space, or it could be false in every Banach space that is not isomorphic to a Hilbert space. $\endgroup$
    – Bill Johnson
    Commented Aug 11, 2021 at 0:49
  • 1
    $\begingroup$ @Bill Johnson Have you looked at "Decomposition of Banach Space Operators" KENNETH R. DAVIDSON and DOMINGO A. HERRERO Indiana University Mathematics Journal Vol. 35, No. 2 (Summer, 1986), pp. 333-343 ? $\endgroup$
    – Derek
    Commented Apr 3, 2022 at 22:42
  • $\begingroup$ @Derek.Thanks. I see that Davidson and Herrero prove in Proposition 5.1 that on $L_p(\mu)$, $1 \le p < \infty$, every compact quasi-nilpotent operator is the norm limit of finite rank nilpotent operators. I read this paper a long time ago (actually, before they submitted it--the authors thank me in the introduction), but forgot about Proposition 5.1. $\endgroup$
    – Bill Johnson
    Commented Apr 4, 2022 at 23:01

1 Answer 1

Reset to default
1
$\begingroup$

Halmos's BAMS paper's references include: R. G. Douglas and Carl Pearcy, A note on quasitriangular operators, Duke Math. J. 37 (1970), 177–188.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .