Extremal extensions for the sum of nonnegative selfadjoint relations

Hassi, Seppo; Sandovici, Adrian; de Snoo, Henk; Winkler, Henrik

doi:10.1090/S0002-9939-07-08827-2

Extremal extensions for the sum of nonnegative selfadjoint relations
HTML articles powered by AMS MathViewer

by Seppo Hassi, Adrian Sandovici, Henk de Snoo and Henrik Winkler PDF

Proc. Amer. Math. Soc. 135 (2007), 3193-3204 Request permission

Abstract:

The sum $A+B$ of two nonnegative selfadjoint relations (multi-valued operators) $A$ and $B$ is a nonnegative relation. The class of all extremal extensions of the sum $A+B$ is characterized as products of relations via an auxiliary Hilbert space associated with $A$ and $B$. The so-called form sum extension of $A+B$ is a nonnegative selfadjoint extension, which is constructed via a closed quadratic form associated with $A$ and $B$. Its connection to the class of extremal extensions is investigated and a criterion for its extremality is established, involving a nontrivial dependence on $A$ and $B$.

References

Yu. M. Arlinskiĭ, S. Hassi, Z. Sebestyén, and H. S. V. de Snoo, On the class of extremal extensions of a nonnegative operator, Recent advances in operator theory and related topics (Szeged, 1999) Oper. Theory Adv. Appl., vol. 127, Birkhäuser, Basel, 2001, pp. 41–81. MR 1902794
Yu. M. Arlinskiĭ and È. R. Tsekanovskiĭ, Quasiselfadjoint contractive extensions of a Hermitian contraction, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 50 (1988), 9–16, i (Russian); English transl., J. Soviet Math. 49 (1990), no. 6, 1241–1247. MR 975668, DOI 10.1007/BF02209165
Earl A. Coddington, Extension theory of formally normal and symmetric subspaces, Memoirs of the American Mathematical Society, No. 134, American Mathematical Society, Providence, R.I., 1973. MR 0477855
Bálint Farkas and Máté Matolcsi, Commutation properties of the form sum of positive, symmetric operators, Acta Sci. Math. (Szeged) 67 (2001), no. 3-4, 777–790. MR 1876466
B. Farkas and M. Matolcsi, Positive forms on Banach spaces, Acta Math. Hungar. 99 (2003), no. 1-2, 43–55. MR 1973084, DOI 10.1023/A:1024501211008
P. A. Fillmore and J. P. Williams, On operator ranges, Advances in Math. 7 (1971), 254–281. MR 293441, DOI 10.1016/S0001-8708(71)80006-3
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.
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. MR 2188974, DOI 10.1007/s10474-006-0036-6
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, to appear.
Zoltán Sebestyén and Jan Stochel, Restrictions of positive selfadjoint operators, Acta Sci. Math. (Szeged) 55 (1991), no. 1-2, 149–154. MR 1124953
Z. Sebestyén and J. Stochel, On products of unbounded operators, Acta Math. Hungar. 100 (2003), no. 1-2, 105–129. MR 1984863, DOI 10.1023/A:1024660318703