Integral Equations and Operator Theory
Mathematics & Statistics
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).
Loreaux, Jireh and Kaftal, Victor, "Kadison's Pythagorean Theorem and Essential Codimension" (2017). SIUE Faculty Research, Scholarship, and Creative Activity. 118.