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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Almost every quasinilpotent Hilbert space operator is a universal quasinilpotent

Author: Domingo A. Herrero
Journal: Proc. Amer. Math. Soc. 71 (1978), 212-216
MSC: Primary 47A65; Secondary 47B05
MathSciNet review: 500492
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Abstract: Let Q be a quasinilpotent operator acting on a complex separable infinite dimensional Hilbert space; then either ${Q^k}$ is compact for some positive integer k, or the closure of the similarity orbit of Q contains every quasinilpotent operator. Analogous results are shown to be true for the Calkin algebra and for nonseparable Hilbert spaces. For the nonseparable case, the analogous result is true for the closed bilateral ideal $\mathcal {J}$, strictly larger than the ideal of compact operators, if and only if $\mathcal {J}$ is not the ideal associated with an ${\aleph _0}$-regular limit cardinal. For the ideal of compact operators, the problem remains open.

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Keywords: Nilpotent operator, quasinilpotent operator, similarity orbit, universal quasinilpotent operator, compact operator, Calkin algebra, Voiculescu’s noncommutative version of Weyl-von Neumann theorem, representation of <!– MATH ${C^ \ast }$ –> <IMG WIDTH="31" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${C^ \ast }$">-algebras, quasitriangular operator, nonseparable Hilbert space, <!– MATH ${\aleph _0}$ –> <IMG WIDTH="27" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="images/img2.gif" ALT="${\aleph _0}$">-irregular cardinal
Article copyright: © Copyright 1978 American Mathematical Society