# Institute of Mathematics

## A remark concerning Putinar's model of hyponormal weighted shifts

#### Vasile Lauric

###### Received March 21, 2017.   First published December 13, 2017.

Abstract:  The question whether a hyponormal weighted shift with trace class self-commutator is the compression modulo the Hilbert-Schmidt class of a normal operator, remains open. It is natural to ask whether Putinar's construction through which he proved that hyponormal operators are subscalar operators provides the answer to the above question. We show that the normal extension provided by Putinar's theory does not lead to the extension that would provide a positive answer to the question.
Keywords:  weighted shift operator; almost normal operator; hyponormal operator
Classification MSC:  47B20, 47B37
DOI:  10.21136/CMJ.2017.0129-17

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] V. Lauric: Remarks on hyponormal operators and almost normal operators. Matematiche (Catania) 72 (2017), 3-8. MR 3666546
[2] C. Pasnicu: Weighted shifts as direct summands mod $\mathcal C_2$ of normal operators. Dilation Theory, Toeplitz operators, and Other Topics. 7th Int. Conf. Oper. Theory, Timisoara, 1982, Oper. Theory, Adv. Appl. 11 (1983), 275-281. MR 0789643 | Zbl 0527.47021
[3] M. Putinar: Hyponormal operators are subscalar. J. Oper. Theory 12 (1984), 385-395. MR 0757441 | Zbl 0573.47016
[4] D. Voiculescu: Hilbert space operators modulo normed ideals. Proc. Int. Congr. Math. Warszawa, 1983 2 (1984), 1041-1047. MR 0804756 | Zbl 0594.46063
[5] D. V. Voiculescu: Almost normal operators mod Hilbert-Schmidt and the $K$-theory of the Banach algebras $E\Lambda(\Omega)$. J. Noncommut. Geom. 8 (2014), 1123-1145. DOI 10.4171/JNCG/181 | MR 3310942 | Zbl 1325.46074

Affiliations:   Vasile Lauric, Department of Mathematics, Florida A&M University, 1617 S Martin Luther King Jr. Blvd., 308 Jackson-Davis Hall, Tallahassee, Florida 32307, USA, e-mail: vasile.lauric@famu.edu

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