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Kadison’s Pythagorean Theorem and Essential Codimension

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Abstract

Kadison’s Pythagorean theorem (Proc Natl Acad Sci USA, 99(7):4178–4184, 2002; Proc Natl Acad Sci USA 99(8):5217–5222, 2002) provides a characterization of the diagonals of projections with a subtle integrality condition. Arveson (Proc Natl Acad Sci USA 104(4):1152–1158, 2007), Kaftal, Ng, Zhang (J Funct Anal 257(8):2497–2529, 2009), and Argerami (Integral Equ Oper Theory 82(1):33–49, 2015) all provide different proofs of that integrality condition. In this paper we interpret the integrality condition in terms of the essential codimension of a pair of projections introduced by Brown et al. (Proceedings of a conference on operator theory, Lecture notes in mathematics, Springer, Berlin, 1973), or, equivalently of the index of a Fredholm pair of projections introduced by Avron, Seiler and Simon (J Funct Anal 120(1):220–237, 1994). The same techniques explain the integer occurring in the characterization of diagonals of selfadjoint operators with finite spectrum by Bownik and Jasper (Trans Am Math Soc 367(7):5099–5140, 2015).

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Authors and Affiliations

  1. Department of Mathematics, University of Cincinnati, P. O. Box 210025 , Cincinnati, OH, 45221-0025, USA

    Victor Kaftal

  2. Department of Mathematics and Statistics, Southern Illinois University Edwardsville, 1 Hairpin Dr, Edwardsville, IL, 62026-1653, USA

    Jireh Loreaux

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  1. Victor Kaftal
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  2. Jireh Loreaux
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Correspondence to Jireh Loreaux.

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This work was partially supported by the Simons Foundation Grant No. 245660 to Victor Kaftal.

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Kaftal, V., Loreaux, J. Kadison’s Pythagorean Theorem and Essential Codimension. Integr. Equ. Oper. Theory 87, 565–580 (2017). https://doi.org/10.1007/s00020-017-2365-y

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  • DOI: https://doi.org/10.1007/s00020-017-2365-y

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