Frames and Finite-Rank Integral Representations of Positive Operator-Valued Measures

  • Jean-Pierre GabardoEmail author
  • Deguang Han


Discrete and continuous frames can be considered as positive operator-valued measures (POVMs) that have integral representations using rank-one operators. However, not every POVM has an integral representation. One goal of this paper is to examine the POVMs that have finite-rank integral representations. More precisely, we present a necessary and sufficient condition under which a positive operator-valued measure \(F: \varOmega \to B(H)\) has an integral representation of the form
$$ F(E) =\sum_{k=1}^{m} \int _{E} G_{k}(\omega )\otimes G_{k}(\omega )\, d \mu (\omega ) $$
for some weakly measurable maps \(G_{k}\ (1\leq k\leq m) \) from a measurable space \(\varOmega \) to a Hilbert space ℋ and some positive measure \(\mu \) on \(\varOmega \). Similar characterizations are also obtained for projection-valued measures. As special consequences of our characterization we settle negatively a problem of Ehler and Okoudjou about probability frame representations of probability POVMs, and prove that an integral representable probability POVM can be dilated to a integral representable projection-valued measure if and only if the corresponding measure is purely atomic.


Positive operator valued measures Frames Integral representations Frames Dilations 

Mathematics Subject Classification

42C15 46C05 47B10 



The authors would like to thank the referees to carefully reading the manuscript and giving helpful comments that help to improve the quality of the paper.


  1. 1.
    Diestel, J., Jarchow, H., Tonge Andrew, A.: Absolutely Summing Operators. Cambridge Studies in Advanced Mathematics, vol. 43. Cambridge University Press, Cambridge (1995) CrossRefzbMATHGoogle Scholar
  2. 2.
    Ehler, M., Okoudjou, K.: Probabilistic frames: an overview. In: Finite Frames, pp. 415–436. Springer, New York (2013) CrossRefGoogle Scholar
  3. 3.
    Gupta, V., Mandayam, P., Sunder, V.S.: A Collection of Notes Based on Lectures by Gilles Pisier, K.R. Parthasarathy, Vern Paulsen and Andreas Winter. Lecture Notes in Physics, vol. 902. Springer, Cham (2015) zbMATHGoogle Scholar
  4. 4.
    Halmos, P.R.: A Hilbert Space Problem Book, 2nd edn. Graduate Texts in Mathematics, vol. 19. Springer, New York/Berlin (1982). Encyclopedia of Mathematics and its Applications, vol. 17 CrossRefzbMATHGoogle Scholar
  5. 5.
    Han, D., Larson, D.: Frames, bases and group representations. Mem. Am. Math. Soc. 147, 697 (2000) MathSciNetzbMATHGoogle Scholar
  6. 6.
    Han, D., Larson, D., Liu, B., Liu, R.: Operator-valued measures, dilations, and the theory of frames. Mem. Am. Math. Soc. 229, 1075 (2014) MathSciNetzbMATHGoogle Scholar
  7. 7.
    Han, D., Larson, D., Liu, B., Liu, R.: Dilations for systems of imprimitivity acting on Banach spaces. J. Funct. Anal. 266(12), 6914–6937 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Han, D., Larson, D., Liu, R.: Dilations of operator-valued measures with bounded \(p\)-variations and framings on Banach spaces. J. Funct. Anal. 274(5), 1466–1490 (2018) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Heinosaari, T., Mazzarella, L., Wolf, M.: Quantum tomography and prior information. Commun. Math. Phys. 318(2), 355–374 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hytönen, T., Pellonpää, J., Ylinen, K.: Positive sesquilinear form measures and generalized eigenvalue expansions. J. Math. Anal. Appl. 336(2), 1287–1304 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Maslouhi, M., Loukili, S.: Probabilistic tight frames and representation of positive operator-valued measures. Appl. Comput. Harmon. Anal. (2018, to appear). Google Scholar
  12. 12.
    Moran, B., Howard, S., Cochran, D.: Positive-operator-valued measures: a general setting for frames. In: Excursions in Harmonic Analysis, vol. 2, pp. 49–64. Springer, New York (2013) CrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada
  2. 2.Department of MathematicsUniversity of Central FloridaOrlandoUSA

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