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Let A be a quasinilpotent operator on a Hilbert space and let every operator of the algebra generated by $A$ and $A^{*}$ have finite spectrum. Does then follow, that A is nilpotent ?

See also quasinilpotence and finite spectrum and Finite-dimensional subalgebras of $C^\star$-algebras

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Do you mean the (norm-)closed algebra generated by $A$ and $A*$? –  Yemon Choi Oct 15 '11 at 1:50
@Yemon Choi : no, not the norm-closed algebra. I mean the non-commutative polynomials in A and $A^{*}$ . –  jjcale Oct 15 '11 at 12:17

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