Banach Journal of Mathematical Analysis

Factorized sectorial relations, their maximal-sectorial extensions, and form sums

Seppo Hassi, Adrian Sandovici, and Henk de Snoo

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In this paper we consider sectorial operators, or more generally, sectorial relations and their maximal-sectorial extensions in a Hilbert space H. Our particular interest is in sectorial relations S, which can be expressed in the factorized form S=T(I+iB)TorS=T(I+iB)T, where B is a bounded self-adjoint operator in a Hilbert space K and T:HK (or T:KH, respectively) is a linear operator or a linear relation which is not assumed to be closed. Using the specific factorized form of S, a description of all the maximal-sectorial extensions of S is given, along with a straightforward construction of the extreme extensions SF, the Friedrichs extension, and SK, the Kreĭn extension of S, which uses the above factorized form of S. As an application of this construction, we also treat the form sum of maximal-sectorial extensions of two sectorial relations.

Article information

Banach J. Math. Anal., Volume 13, Number 3 (2019), 538-564.

Received: 23 October 2018
Accepted: 28 January 2019
First available in Project Euclid: 25 May 2019

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Zentralblatt MATH identifier

Primary: 47B44: Accretive operators, dissipative operators, etc.
Secondary: 47A06: Linear relations (multivalued linear operators) 47A07: Forms (bilinear, sesquilinear, multilinear) 47B65: Positive operators and order-bounded operators

sectorial relation Friedrichs extension Kreĭn extension extremal extension form sum


Hassi, Seppo; Sandovici, Adrian; de Snoo, Henk. Factorized sectorial relations, their maximal-sectorial extensions, and form sums. Banach J. Math. Anal. 13 (2019), no. 3, 538--564. doi:10.1215/17358787-2019-0003.

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  • [1] T. Ando and K. Nishio, Positive selfadjoint extensions of positive symmetric operators, Tohoku Math. J. (2) 22 (1970), 65–75.
  • [2] Y. M. Arlinskiĭ, Maximal sectorial extensions and closed form associated with them, Ukrainian Math. J. 48 (1996), no. 6, 809–827.
  • [3] Y. M. Arlinskiĭ, Extremal extensions of sectorial linear relations, Mat. Stud. 7 (1997), no. 1, 81–96.
  • [4] Y. M. Arlinskiĭ, “Boundary triplets and maximal accretive extensions of sectorial operators” in Operator Methods for Boundary Value Problems, London Math. Soc. Lecture Note Ser. 404, Cambridge Univ. Press, Cambridge, 2012, 35–72.
  • [5] Y. M. Arlinskiĭ, S. Hassi, Z. Sebestyén, and H. S. V. de Snoo, “On the class of extremal extensions of a nonnegative operator” in Recent Advances in Operator Theory and Related Topics (Szeged, 1999), Birkhäuser, Basel, 2001, 41–81.
  • [6] E. A. Coddington, Extension Theory of Formally Normal and Symmetric Subspaces, Mem. Amer. Math. Soc. 134, Amer. Math. Soc., Providence, 1973.
  • [7] E. A. Coddington and H. S. V. de Snoo, Positive selfadjoint extensions of positive symmetric subspaces, Math. Z. 159 (1978), 203–214.
  • [8] R. G. Douglas, On majorization, factorization and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413–415.
  • [9] B. Farkas and M. Matolcsi, Commutation properties of the form sum of positive symmetric operators, Acta Sci. Math. (Szeged) 67 (2001), no. 3–4, 777–790.
  • [10] S. Hassi, “On the Friedrichs and the Kreĭn-von Neumann extension of nonnegative relations” in Contributions to Management Science, Mathematics and Modelling, Universitas Wasaensis, Vaasa, 2004, 37–54.
  • [11] S. Hassi and H. S. V. de Snoo, A class of sectorial relations and the associated closed forms, in preparation.
  • [12] S. Hassi, M. M. Malamud, and H. S. V. de Snoo, On Kreĭn’s extension theory of nonnegative operators, Math. Nachr. 274/275 (2004), 40–73.
  • [13] S. Hassi, A. Sandovici, H. S. V. de Snoo, and H. Winkler, Form sums of nonnegative selfadjoint operators, Acta Math. Hungar. 111 (2006), no. 1–2, 81–105.
  • [14] S. Hassi, A. Sandovici, H. S. V. de Snoo, and H. Winkler, Extremal extensions for the sum of nonnegative selfadjoint relations, Proc. Amer. Math. Soc. 135 (2007), no. 10, 3193–3204.
  • [15] S. Hassi, A. Sandovici, H. S. V. de Snoo, and H. Winkler, A general factorization approach to the extension theory of nonnegative operators and relations, J. Operator Theory 58 (2007), no. 2, 351–386.
  • [16] S. Hassi, A. Sandovici, H. S. V. de Snoo, and H. Winkler, Extremal maximal sectorial extensions of sectorial relations, Indag. Math. (N.S.) 28 (2017), no. 5, 1019 –1055.
  • [17] S. Hassi, Z. Sebestyén, and H. S. V. de Snoo, Lebesgue type decompositions for linear relations and Ando’s uniqueness criterion, Acta Sci. Math. (Szeged) 84 (2018), no. 3–4, 465–507.
  • [18] T. Kato, Perturbation Theory for Linear Operators, Grundlehren Math. Wiss. 132, Springer, Berlin, 1980.
  • [19] M. G. Kreĭn, Theory of selfadjoint extensions of semibounded operators and its applications I, II, Mat. Sb. 20, 21 (1947), 431–495, 365–404.
  • [20] V. Prokaj and Z. Sebestyén, On Friedrichs extensions of operators, Acta Sci. Math. (Szeged) 62 (1996), no. 1–2, 243–246.
  • [21] F. S. Rofe-Beketov, The numerical range of a linear relation and maximum relations, Funktsional. Anal. i Prilozhen 44 (1985), 103–112 (Russian), English translation in J. Soviet Math. 48 (1990), no. 3, 329–336.
  • [22] Z. Sebestyén and E. Sikolya, On Kreĭn-von Neumann and Friedrichs extensions, Acta Sci. Math. (Szeged) 69 (2003), no. 1–2, 323–336.
  • [23] Z. Sebestyén and J. Stochel, Restrictions of positive self-adjoint operators, Acta Sci. Math. (Szeged) 55 (1991), no. 1–2, 149–154.
  • [24] Z. Sebestyén and Z. Tarcsay, $T^{*}T$ always has a positive selfadjoint extension, Acta Math. Hungar. 135 (2012), 116–129.