zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The many proofs of an identity on the norm of oblique projections. (English) Zbl 1102.47002
Summary: 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=I-P. This useful equality, while not widely-known, has been proven repeatedly in the literature. Many published proofs are reviewed, and simpler ones are presented.
47A05General theory of linear operators
15A24Matrix equations and identities
15A60Applications of functional analysis to matrix theory
46C99Inner product spaces, Hilbert spaces
46E99Linear function spaces and their duals
46N40Applications of functional analysis in numerical analysis
[1]Afriat, S.N.: Orthogonal and oblique projectors and the characteristics of pairs of vector spaces. Proc. Camb. Philos. Soc. 53, 800–816 (1957) · doi:10.1017/S0305004100032916
[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)
[4]Beattie, C.: Galerkin eigenvector approximations. Math. Comput. 69, 1409–1434 (2000) · Zbl 0956.65043 · doi:10.1090/S0025-5718-00-01181-9
[5]Beattie, C., Embree, M., Rossi, J.: Convergence of restarted Krylov subspaces to invariant subspaces. SIAM J. Matrix Anal. Appl. 25, 1074–1109 (2004) · Zbl 1067.65037 · doi:10.1137/S0895479801398608
[6]Björck, Å., Golub, G.H.: Numerical methods for computing angles between linear subspaces. Math. Comput. 27, 579–594 (1973)
[7]Buckholtz, D.: Hilbert space idempotents and involutions. Proc. Am. Math. Soc. 128, 1415–1418 (1999) · Zbl 0955.46015 · doi:10.1090/S0002-9939-99-05233-8
[8]Chatelin, F.: Spectral Approximation of Linear Operators. Academic, New York (1983)
[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)
[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)
[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)
[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) · Zbl 1099.46012 · doi:10.4064/sm170-2-6
[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)
[14]Drmač, Z.: On principal angles between subspaces of Euclidean space. SIAM J. Matrix Anal. Appl. 22, 173–194 (2000) · Zbl 0968.65021 · doi:10.1137/S0895479897320824
[15]Eiermann, M., Ernst, O.: Geometric aspects in the theory of Krylov subspace methods. Acta Numer. 10, 251–312 (2001) · Zbl 1105.65328 · doi:10.1017/S0962492901000046
[16]Falgout, R.D., Vassilevski, P.S., Zikatanov, L.T.: On two-grid convergence estimates. Numer. Linear Algebra Appl. 12, 471–494 (2005) · Zbl 1164.65343 · doi:10.1002/nla.437
[17]Galantái, A., Hegedüs, C.J.: Jordan’s principal angles in complex vector spaces. Numer. Linear Algebra Appl. 13, 589–598 (2006) · Zbl 1174.15300 · doi:10.1002/nla.491
[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)
[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)
[20]Gurarĭi, V.I.: Openings and inclinations of subspaces of a Banach space. Teor. Funkc. Funkc. Anal. ih Priloz. 1, 194–204 (1965)
[21]Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. John Hopkins University Press, Baltimore (1996)
[22]Ipsen, I.C.F., Meyer, C.D.: The angle between complementary subspaces. Am. Math. Mon. 102, 904–911 (1995) · Zbl 0842.15002 · doi:10.2307/2975268
[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) · Zbl 0119.32001 · doi:10.1007/BF01386205
[25]Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin Heidelberg New York (1980)
[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) · Zbl 1018.65058 · doi:10.1137/S1064827500377332
[27]Koliha, J.J.: Range projections of idempotents in C *-algebras. Demonstr. Math. 34, 91–103 (2001)
[28]Koliha, J.J., Rakočević, V.: On the norm of idempotents in C *-algebras. Rocky Mt. J. Math. 34, 685–697 (2004) · Zbl 1066.46044 · doi:10.1216/rmjm/1181069874
[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)
[31]Lewkowitcz, I.: Bounds for the singular values of a matrix with nonnegative eigenvalues. Linear Algebra Appl. 112, 29–37 (1989) · Zbl 0659.15017 · doi:10.1016/0024-3795(89)90586-7
[32]Ljance, V.É.: Some properties of idempotent operators. Teor. Prikl. Matematica 1, 16–22 (1958/1959)
[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) · Zbl 1127.76049 · doi:10.1016/j.apnum.2004.09.022
[34]Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia (2000)
[35]Pták, V.: Extremal operators and oblique projections. Čas. Pěst. Mat. 110, 343–350 (1985)
[36]Rakočević, V.: On the norm of idempotent operators in a Hilbert space. Am. Math. Mon. 107, 748–750 (2000) · Zbl 0993.47009 · doi:10.2307/2695474
[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) · Zbl 1079.65034 · doi:10.1137/S0036144503424439
[38]Steinberg, J.: Oblique projections in Hilbert spaces. Integr. Equ. Oper. Theory 38, 81–119 (2000) · Zbl 0987.46027 · doi:10.1007/BF01192303
[39]Stewart, G.W., Sun, J.-G.: Matrix Perturbation Theory. Academic, San Diego, California, USA (1990)
[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)
[41]Wimmer, H.K.: Canonical angles of unitary spaces and perturbations of direct complements. Linear Algebra Appl. 287, 373–379 (1999) · Zbl 0937.15002 · doi:10.1016/S0024-3795(98)10017-4
[42]Wimmer, H.K.: Lipschitz continuity of oblique projections. Proc. Am. Math. Soc. 128, 873–876 (1999) · Zbl 0935.46014 · doi:10.1090/S0002-9939-99-05267-3
[43]Xu, J., Zikatanov, L.: Some observations on Babuška and Brezzi theories. Numer. Math. 94, 195–202 (2003) · Zbl 1028.65115 · doi:10.1007/s002110100308