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The many proofs of an identity on the norm of oblique projections

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Abstract

Given an oblique projector P on a Hilbert space, i.e., an operator satisfying P 2=P, which is neither null nor the identity, it holds that ||P|| = ||IP||. This useful equality, while not widely-known, has been proven repeatedly in the literature. Many published proofs are reviewed, and simpler ones are presented.

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References

  1. Afriat, S.N.: Orthogonal and oblique projectors and the characteristics of pairs of vector spaces. Proc. Camb. Philos. Soc. 53, 800–816 (1957)

    Article  MathSciNet  Google Scholar 

  2. Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. F. Ungar, New York (1961 and 1963) (Reprinted by Dover, New York (1993))

  3. Andruchov, E., Corach, G.: Geometry of oblique projections. Stud. Math. 137, 61–79 (1999)

    Google Scholar 

  4. Beattie, C.: Galerkin eigenvector approximations. Math. Comput. 69, 1409–1434 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Beattie, C., Embree, M., Rossi, J.: Convergence of restarted Krylov subspaces to invariant subspaces. SIAM J. Matrix Anal. Appl. 25, 1074–1109 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Björck, Å., Golub, G.H.: Numerical methods for computing angles between linear subspaces. Math. Comput. 27, 579–594 (1973)

    Article  MATH  Google Scholar 

  7. Buckholtz, D.: Hilbert space idempotents and involutions. Proc. Am. Math. Soc. 128, 1415–1418 (1999)

    Article  MathSciNet  Google Scholar 

  8. Chatelin, F.: Spectral Approximation of Linear Operators. Academic, New York (1983)

    MATH  Google Scholar 

  9. Corach, G., Maestripieri, A., Stojanoff, D.: A classification of projectors. In: Jarosz, K., Sołtysiak, A. (eds.) Topological Algebras, their Applications, and Related Topics. Banach Center Publications, vol. 67, pp. 145–160. Polish Academy of Sciences, Warsaw (2005)

    Google Scholar 

  10. Del Pasqua, D.: Su una nozione di varietà lineari disgiunte di uno spazio di Banach (On a notion of disjoint linear manifolds of a Banach space). Rend. Mat. Appl. 5, 406–422 (1955)

    Google Scholar 

  11. Deutsch, F.: The angle between subspaces of a Hilbert space. In: Singh, S.P. (ed.) Approximation Theory, Wavelets and Applications, Proceedings of the NATO Advanced Study Institute with the assistance of Antonio Carbone and B. Watson, pp. 107–130. Kluwer, Dordrecht, The Netherlands (1995)

    Google Scholar 

  12. Dirr, G., Rakočević, V., Wimmer, H.K.: Estimates for projections in Banach spaces and existence of direct complements. Stud. Math. 170, 211–216 (2005)

    Article  MATH  Google Scholar 

  13. Dixmier, J.: Étude sur les variétés et les opérateurs de Julia, avec quelques applications (On Julia varieties and operators with applications). Bull. Soc. Math. Fr. 77, 11–101 (1949)

    MATH  MathSciNet  Google Scholar 

  14. Drmač, Z.: On principal angles between subspaces of Euclidean space. SIAM J. Matrix Anal. Appl. 22, 173–194 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Eiermann, M., Ernst, O.: Geometric aspects in the theory of Krylov subspace methods. Acta Numer. 10, 251–312 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Falgout, R.D., Vassilevski, P.S., Zikatanov, L.T.: On two-grid convergence estimates. Numer. Linear Algebra Appl. 12, 471–494 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Galantái, A., Hegedüs, C.J.: Jordan’s principal angles in complex vector spaces. Numer. Linear Algebra Appl. 13, 589–598 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gohberg, I.C., Kreǐn, M.G.: Introduction to the theory of nonselfadjoint operators. Nauka, Moscow, 1965. Translations of Mathematical Monographs, vol. 18, American Mathematical Society, Providence, Rhode Island (1969)

    Google Scholar 

  19. Gohberg, I.C., Lancaster, P., Rodman, L.: Invariant Subspaces of Matrices with Applications. Wiley, New York (1986) (Reprinted by SIAM, 2006. Classics in Applied Mathematics, vol. 51)

    MATH  Google Scholar 

  20. Gurarĭi, V.I.: Openings and inclinations of subspaces of a Banach space. Teor. Funkc. Funkc. Anal. ih Priloz. 1, 194–204 (1965)

    Google Scholar 

  21. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. John Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  22. Ipsen, I.C.F., Meyer, C.D.: The angle between complementary subspaces. Am. Math. Mon. 102, 904–911 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  23. Jujunashvili, A.: Angles between infinite-dimensional subspaces. PhD thesis, Department of Applied Mathematics, University of Colorado, Denver (2005)

  24. Kato, T.: Estimation of iterated matrices, with application to the von Neumann condition. Numer. Math. 2, 22–29 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin Heidelberg New York (1980)

    MATH  Google Scholar 

  26. Knyazev, A., Argentati, M.E.: Principal angles between subspaces in an A-based scalar product: algorithms and perturbation estimates. SIAM J. Sci. Comput. 23, 2008–2040 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  27. Koliha, J.J.: Range projections of idempotents in C *-algebras. Demonstr. Math. 34, 91–103 (2001)

    MATH  MathSciNet  Google Scholar 

  28. Koliha, J.J., Rakočević, V.: On the norm of idempotents in C *-algebras. Rocky Mt. J. Math. 34, 685–697 (2004)

    MATH  Google Scholar 

  29. Kreǐn, M.G., Krasnolsel’skiǐ, M.A., Mil’man, D.P.: On the defect numbers of linear operators in a Banach space and on some geometric questions. In: Akademiya Nauk Ukrainskoĭ RSR. Zbirnik Prac’ Insttitut Matematiki. (Collection of Papers of the Institute of Mathematics of the Academy of Sciences of Ukraine), vol. 11, pp. 97–112 (1948)

  30. Labrousse, J.-P.: Une caracterization topologique des générateurs infinitésimaux de semi-groupes analytiques et de contractions sur un espace de Hilbert (A topological characterization of the infinitesimal generators of analytic semigroups and of contractions on a Hilbert space). Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., Rend. Lincei Suppl. 52, 631–636 (1972)

    MATH  MathSciNet  Google Scholar 

  31. Lewkowitcz, I.: Bounds for the singular values of a matrix with nonnegative eigenvalues. Linear Algebra Appl. 112, 29–37 (1989)

    Article  MathSciNet  Google Scholar 

  32. Ljance, V.É.: Some properties of idempotent operators. Teor. Prikl. Matematica 1, 16–22 (1958/1959)

    MathSciNet  Google Scholar 

  33. Mandel, J., Dohrmann, C.R., Tezaur, R.: An algebraic theory for primal and dual substructuring methods by constraints. Appl. Numer. Math. 54, 167–193 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  34. Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia (2000)

    MATH  Google Scholar 

  35. Pták, V.: Extremal operators and oblique projections. Čas. Pěst. Mat. 110, 343–350 (1985)

    MATH  Google Scholar 

  36. Rakočević, V.: On the norm of idempotent operators in a Hilbert space. Am. Math. Mon. 107, 748–750 (2000)

    Article  MATH  Google Scholar 

  37. Simoncini, V., Szyld, D.B.: On the occurrence of superlinear convergence of exact and inexact Krylov subspace methods. SIAM Rev. 47, 247–272 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  38. Steinberg, J.: Oblique projections in Hilbert spaces. Integr. Equ. Oper. Theory 38, 81–119 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  39. Stewart, G.W., Sun, J.-G.: Matrix Perturbation Theory. Academic, San Diego, California, USA (1990)

    MATH  Google Scholar 

  40. Wedin, P.-Å.: On angles between subspaces of a finite dimensional inner product space. In: Kågström, B., Ruhe, A. (eds.) Matrix Pencils, Proceedings of a Conference Held at Pite Havsbad, Sweden, March 22–24, 1982. Lecture Notes in Mathematics, vol. 973, pp. 263–285. Springer, Berlin Heidelberg New York (1983)

    Google Scholar 

  41. Wimmer, H.K.: Canonical angles of unitary spaces and perturbations of direct complements. Linear Algebra Appl. 287, 373–379 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  42. Wimmer, H.K.: Lipschitz continuity of oblique projections. Proc. Am. Math. Soc. 128, 873–876 (1999)

    Article  MathSciNet  Google Scholar 

  43. Xu, J., Zikatanov, L.: Some observations on Babuška and Brezzi theories. Numer. Math. 94, 195–202 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Daniel B. Szyld.

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Communicated by Michele Benzi.

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Szyld, D.B. The many proofs of an identity on the norm of oblique projections. Numer Algor 42, 309–323 (2006). https://doi.org/10.1007/s11075-006-9046-2

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