Sums of products of positive operators and spectra of Lüders operators

Magajna, Bojan

doi:10.1090/S0002-9939-2012-11537-0

Sums of products of positive operators and spectra of Lüders operators
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Proc. Amer. Math. Soc. 141 (2013), 1349-1360 Request permission

Abstract:

Each bounded operator $T$ on an infinite dimensional Hilbert space $\mathcal {H}$ is a sum of three operators that are similar to positive operators; two such operators are sufficient if $T$ is not a compact perturbation of a scalar. The spectra of Lüders operators (elementary operators on $\mathrm {B}(\mathcal {H})$ with positive coefficients) of lengths at least three are not necessarily contained in $\mathrm {B}(\mathcal {H})^+$. On the other hand, the spectra of such operators of lengths (at most) two are contained in $\mathrm {B}(\mathcal {H})^+$ if the coefficients on one side commute.

References

J. H. Anderson and J. G. Stampfli, Commutators and compressions, Israel J. Math. 10 (1971), 433–441. MR 312312, DOI 10.1007/BF02771730
Arlen Brown and Carl Pearcy, Structure of commutators of operators, Ann. of Math. (2) 82 (1965), 112–127. MR 178354, DOI 10.2307/1970564
John B. Conway, A course in functional analysis, Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1985. MR 768926, DOI 10.1007/978-1-4757-3828-5
John B. Conway, A course in operator theory, Graduate Studies in Mathematics, vol. 21, American Mathematical Society, Providence, RI, 2000. MR 1721402, DOI 10.1090/gsm/021
Raúl E. Curto, Spectral theory of elementary operators, Elementary operators and applications (Blaubeuren, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 3–52. MR 1183936
Kenneth R. Davidson and Laurent W. Marcoux, Linear spans of unitary and similarity orbits of a Hilbert space operator, J. Operator Theory 52 (2004), no. 1, 113–132. MR 2091463
Peng Fan, Che Kao Fong, and Domingo A. Herrero, On zero-diagonal operators and traces, Proc. Amer. Math. Soc. 99 (1987), no. 3, 445–451. MR 875378, DOI 10.1090/S0002-9939-1987-0875378-9
David W. Kribs, A quantum computing primer for operator theorists, Linear Algebra Appl. 400 (2005), 147–167. MR 2131922, DOI 10.1016/j.laa.2004.11.010
Long Long and Shifang Zhang, Fixed points of commutative super-operators, J. Phys. A 44 (2011), no. 9, 095201, 10. MR 2771865, DOI 10.1088/1751-8113/44/9/095201
Bojan Magajna, Fixed points of normal completely positive maps on $\textrm {B}(\scr H)$, J. Math. Anal. Appl. 389 (2012), no. 2, 1291–1302. MR 2879297, DOI 10.1016/j.jmaa.2012.01.007
Gabriel Nagy, On spectra of Lüders operations, J. Math. Phys. 49 (2008), no. 2, 022110, 8. MR 2392847, DOI 10.1063/1.2840472
Carl Pearcy and David Topping, Sums of small numbers of idempotents, Michigan Math. J. 14 (1967), 453–465. MR 218922
Bebe Prunaru, Fixed points for Lüders operations and commutators, J. Phys. A 44 (2011), no. 18, 185203, 2. MR 2788723, DOI 10.1088/1751-8113/44/18/185203
Heydar Radjavi and Peter Rosenthal, Invariant subspaces, 2nd ed., Dover Publications, Inc., Mineola, NY, 2003. MR 2003221
V. S. Shulman and Yu. V. Turovskii, Topological radicals, II. Applications to spectral theory of multiplication operators, Elementary operators and their applications, Oper. Theory Adv. Appl., vol. 212, Birkhäuser/Springer Basel AG, Basel, 2011, pp. 45–114. MR 2789133, DOI 10.1007/978-3-0348-0037-2_{5}
Joseph G. Stampfli, Sums of projections, Duke Math. J. 31 (1964), 455–461. MR 165376
Dan Voiculescu, Some results on norm-ideal perturbations of Hilbert space operators, J. Operator Theory 2 (1979), no. 1, 3–37. MR 553861
Liu Weihua and Wu Junde, Fixed points of commutative Lüders operations, J. Phys. A 43 (2010), no. 39, 395206, 9. MR 2720064, DOI 10.1088/1751-8113/43/39/395206
Pei Yuan Wu, Additive combinations of special operators, Functional analysis and operator theory (Warsaw, 1992) Banach Center Publ., vol. 30, Polish Acad. Sci. Inst. Math., Warsaw, 1994, pp. 337–361. MR 1285620