Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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OPERATORS A, B FOR WHICH THE ALUTHGE TRANSFORM ${\tilde{AB}}$ IS A GENERALISED n-PROJECTION

  • Bhagwati P. Duggal (Faculty of Sciences and Mathematics University of Nis) ;
  • In Hyoun Kim (Department of Mathematics Incheon National University)
  • Received : 2022.10.27
  • Accepted : 2023.02.24
  • Published : 2023.11.30
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Abstract

A Hilbert space operator A ∈ B(H) is a generalised n-projection, denoted A ∈ (G-n-P), if A*n = A. (G-n-P)-operators A are normal operators with finitely countable spectra σ(A), subsets of the set $\{0\}\,{\cup}\,\{\sqrt[n+1]{1}\}.$ The Aluthge transform à of A ∈ B(H) may be (G - n - P) without A being (G - n - P). For doubly commuting operators A, B ∈ B(H) such that σ(AB) = σ(A)σ(B) and ${\parallel}A{\parallel}\,{\parallel}B{\parallel}\;{\leq}\;{\parallel}{\tilde{AB}}{\parallel},$ ${\tilde{AB}}\;{\in}\;(G\,-\,n\,-\,P)$ if and only if $A\;=\;{\parallel}{\tilde{A}}{\parallel}\,(A_{00}\,{\oplus}\,(A_0\,{\oplus}\,A_u))$ and $B\;=\;{\parallel}{\tilde{B}}{\parallel}\,(B_0\,{\oplus}\,B_u),$ where A00 and B0, and A0 ⊕ Au and Bu, doubly commute, A00B0 and A0 are 2 nilpotent, Au and Bu are unitaries, A*nu = Au and B*nu = Bu. Furthermore, a necessary and sufficient condition for the operators αA, βB, αà and ${\beta}{\tilde{B}},\;{\alpha}\,=\,\frac{1}{{\parallel}{\tilde{A}}{\parallel}}$ and ${\beta}\,=\,\frac{1}{{\parallel}{\tilde{B}}{\parallel}},$ to be (G - n - P) is that A and B are spectrally normaloid at 0.

Keywords

Acknowledgement

The second named author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1F1A1057574).

References

  1. J. K. Baksalary and X. J. Liu, An alternative characterization of generalized projectors, Linear Algebra Appl. 388 (2004), 61-65. https://doi.org/10.1016/j.laa.2004.01.010 
  2. B. A. Barnes, Common operator properties of the linear operators RS and SR, Proc. Amer. Math. Soc. 126 (1998), no. 4, 1055-1061. https://doi.org/10.1090/S0002-9939-98-04218-X 
  3. A. Brown and C. M. Pearcy, Spectra of tensor products of operators, Proc. Amer. Math. Soc. 17 (1966), 162-166. https://doi.org/10.2307/2035080 
  4. J. J. Buoni and J. D. Faires, Ascent, descent, nullity and defect of products of operators, Indiana Univ. Math. J. 25 (1976), no. 7, 703-707. https://doi.org/10.1512/iumj.1976.25.25054 
  5. H. K. Du and Y. Li, The spectral characterization of generalized projections, Linear Algebra Appl. 400 (2005), 313-318. https://doi.org/10.1016/j.laa.2004.11.027 
  6. B. P. Duggal and I. H. Kim, Products of generalized n-projections, Linear Multilinear Algebra. https://doi.org/10.1080/03081087.2021.2002249 
  7. J. Gross and G. Trenkler, Generalized and hypergeneralized projectors, Linear Algebra Appl. 264 (1997), 463-474. https://doi.org/10.1016/S0024-3795(96)00541-1 
  8. H. G. Heuser, Functional Analysis, translated from the German by John Horvath, A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1982. 
  9. L. Lebtahi and N. Thome, A note on k-generalized projections, Linear Algebra Appl. 420 (2007), no. 2-3, 572-575. https://doi.org/10.1016/j.laa.2006.08.011