Abstract
For a bounded operator that is not a sum of scalar and compact operators and is similar to a diagonal operator, we prove that it is a linear combination of three idempotents. It is also proved that any self-adjoint diagonal operator is a linear combination of four orthoprojectors with real coefficients.
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REFERENCES
P. Y. Wu, “Additive combinations of special operators,” Funct. Anal. Oper. Theor., 30, 337–361 (1994).
C. Pearcy and D. M. Topping, “Sums of small numbers of idempotents,” Mich. Math. J., 14, 453–465 (1967).
A. Brown, P. R. Halmos, and C. Pearcy, “Commutators of operators on Hilbert space,” Can. J. Math., 17, 695–708 (1965).
V. I. Rabanovych, “On the decomposition of an operator into a sum of four idempotents,” Ukr. Mat. Zh., 56, No.3, 419–424 (2004).
V. Rabanovich, “Every matrix is a linear combination of three idempotents,” Linear Algebra Its Appl., 390, 137–143 (2004).
K. Matsumoto, “Self-adjoint operators as a real sum of 5 projections,” Math. Jpn., 29, 291–294 (1984).
Y. Nakamura, “Any Hermitian matrix is a linear combination of four projections,” Linear Algebra Its Appl., 61, 133–139 (1984).
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Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 3, pp. 388–393, March, 2005.
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Rabanovych, V.I. On the Decomposition of a Diagonal Operator into a Linear Combination of Idempotents or Projectors. Ukr Math J 57, 466–473 (2005). https://doi.org/10.1007/s11253-005-0203-x
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DOI: https://doi.org/10.1007/s11253-005-0203-x