Uniqueness of commuting compact approximations

Authors:
Richard B. Holmes, Bruce E. Scranton and Joseph D. Ward

Journal:
Trans. Amer. Math. Soc. **208** (1975), 330-340

MSC:
Primary 47A65

MathSciNet review:
0380480

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Abstract: Let be an infinite dimensional complex Hilbert space, and let (resp. ) be the algebra of all bounded (resp. compact) linear operators on . It is well known that every has a best approximation from the subspace . The purpose of this paper is to study the uniqueness problem concerning the best approximation of a bounded linear operator by compact operators. Our criterion for selecting a unique representative from the set of best approximants is that the representative should commute with . In particular, many familiar operators are shown to have zero as a unique commuting best approximant.

**[1]**Arlen Brown, P. R. Halmos, and A. L. Shields,*Cesàro operators*, Acta Sci. Math. (Szeged)**26**(1965), 125–137. MR**0187085****[2]**L. A. Coburn,*Weyl’s theorem for nonnormal operators*, Michigan Math. J.**13**(1966), 285–288. MR**0201969****[3]**James A. Deddens and Tin Kin Wong,*The commutant of analytic Toeplitz operators*, Trans. Amer. Math. Soc.**184**(1973), 261–273. MR**0324467**, 10.1090/S0002-9947-1973-0324467-0**[4]**I. C. Gohberg and M. G. Kreĭn,*Introduction to the theory of linear nonself-adjoint operators in Hilbert space*, ``Nauka", Moscow, 1965; English transl., Transl. Math. Monographs, vol. 18, Amer. Math. Soc., Providence, R. I., 1969. MR**36**#3137;**39**#7447.**[5]**Paul R. Halmos,*A Hilbert space problem book*, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR**0208368****[6]**Richard B. Holmes and Bernard R. Kripke,*Best approximation by compact operators*, Indiana Univ. Math. J.**21**(1971/72), 255–263. MR**0296659****[7]**Richard Holmes, Bruce Scranton, and Joseph Ward,*Approximation from the space of compact operators and other 𝑀-ideals*, Duke Math. J.**42**(1975), 259–269. MR**0394301****[8]**V. I. Lomonosov,*Invariant subspaces of the family of operators that commute with a completely continuous operator*, Funkcional. Anal. i Priložen.**7**(1973), no. 3, 55–56 (Russian). MR**0420305****[9]**Roger D. Nussbaum,*The radius of the essential spectrum*, Duke Math. J.**37**(1970), 473–478. MR**0264434****[10]**A. L. Shields and L. J. Wallen,*The commutants of certain Hilbert space operators*, Indiana Univ. Math. J.**20**(1970/1971), 777–788. MR**0287352****[11]**E. C. Titchmarsh,*The theory of functions*, 2nd ed., Oxford Univ. Press, Oxford, 1939.

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DOI:
https://doi.org/10.1090/S0002-9947-1975-0380480-0

Article copyright:
© Copyright 1975
American Mathematical Society