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Tytuł artykułu

Cauchy-Binet type formulas for Fredholm operators

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Języki publikacji
EN
Abstrakty
EN
Suppose A ∈ L(Y ,Z ) , B ∈ L(X ,Y ) are Fredholm operators acting in linear spaces. By referring to the correspondence between Fredholm operators and their determinant systems, we derive the formulas for a determinant system for AB which are expressed via determinant systems for A and B. In our approach, applying results of the theory of determinant systems plays the crucial role and yields Cauchy-Binet type formulas. The formulas are utilized in many branches of applied science and engineering.
Rocznik
Strony
43--54
Opis fizyczny
Bibliogr. 22 poz.
Twórcy
  • Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn Olsztyn, Poland
Bibliografia
  • [1] Leżański T., The Fredholm theory of linear equations in Banach spaces, Stud. Math. 1953, 13, 244-276.
  • [2] Sikorski R., On Leżański's determinants of linear equations in Banach spaces, Stud. Math. 1953, 14, 24-48.
  • [3] Sikorski R., Determinant systems, Stud. Math. 1959, 18, 161-186.
  • [4] Sikorski R., The determinant theory in Banach spaces, Colloq. Math. 1961, 8, 141-198.
  • [5] Buraczewski A., The determinant theory of generalized Fredholm operators, Studia Math. 1963, 22, 265-307.
  • [6] Buraczewski A., Sikorski R., Analytic formulae for determinant systems in Banach spaces, Studia Math. 1980, 67, 85-101.
  • [7] Buraczewski A., Determinant system for composite of generalized Fredholm operators, Studia Math. 1970, 34, 197-207.
  • [8] Marcus M., Minc H., Introduction to Linear Algebra, Dover Publications, New York 1988.
  • [9] Lancaster P., Tismenetsky M., The Theory of Matrices. Computer Science and Applied Mathematics, Academic Press, New York 1985.
  • [10] Shafarevich I.R., Remizov A.O., Linear Algebra and Geometry, Springer-Verlag, Berlin 2012.
  • [11] Brualdi R.A., Schneider H., Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley, Linear Algebra Appl. 1983, 52/53, 769-791.
  • [12] Vishwanathan S.V.N., Smola A.J., Vidal R., Binet-Cauchy kernels on dynamical systems and its application to the analysis of dynamic scenes, Int. J. Comput. Vision 2007, 73(1), 95-119.
  • [13] Konstantopoulos T., A multilinear algebra proof for the Cauchy-Binet formula and a multilinear version of Parseval’s identity, Linear Algebra Appl. 2013, 439 (9), 2651-2658.
  • [14] Karlin S., Rinott Y., A generalized Cauchy-Binet formula and applications to total positivity and majorization, J. Mult. Anal. 1988, 27, 284-299.
  • [15] Shevelev V., Combinatorial minors for matrix functions and their applications, Zeszyty Naukowe Politechniki Śląskiej 2014, seria: Matematyka Stosowana, 4, 5-16.
  • [16] Caracciolo S., Sokal A.D., Sportiello A., Noncommutative determinants, Cauchy-Binet formulae and Capelli-type identities. I. Generalizations of the Capelli and Turnbull identities, Electron. J. Comb. 2009, 16 (1), Research Paper R103, 1-43.
  • [17] Knill O., Cauchy-Binet for pseudo-determinants, Linear Algebra Appl. 2014, 459, 522-547.
  • [18] Ciecierska G., Determinant systems for nuclear perturbations of Fredholm operators in Frechet spaces, PanAmer. Math. J. 2014, 24(1), 1-20.
  • [19] Ciecierska G., Formulas of Fredholm type for Fredholm linear equations in Frechet spaces, Math. Aeterna 2015, 5(5), 945-960.
  • [20] Ciecierska G., Determinant systems method for computing reflexive generalized inverses of products of Fredholm operators, Math. Aeterna 2016, 6(6), 895-906.
  • [21] Ruston A.F., Fredholm theory in Banach spaces, Cambridge Tracts in Math. 86, Cambridge Univ. Press, Cambridge 1986.
  • [22] Ben-Israel A., Greville T.N.E., Generalized Inverses. Theory and Applications, Springer-Verlag, New York 2003.
Uwagi
Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-03678b3b-ef47-4a55-91be-7ac08aa78161
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