Abstract
The main aim of this paper is to provide some range-type criteria for the essentially self-adjointness of symmetric linear relations in real or complex Hilbert spaces. The main tool is a matrix whose entries are certain linear relations.
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References
R. Arens, Operational calculus of linear relations. Pacific J. Math. 11, 9–23 (1961)
E.A. Coddington, H.S.V. de Snoo, Positive self-adjoint extensions of positive symmetric subspaces. Math. Z. 159, 203–214 (1978)
A. Favini, A. Yagi, Degenerate Differential Equations in Banach Spaces (Marcel Dekker, New York, 1999)
S. Hassi, On the Friedrichs and the Kreĭn-von Neumann extension of nonnegative relations. Acta Wasaensia 122, 37–54 (2004)
S. Hassi, A. Sandovici, H.S.V. de Snoo, H. Winkler, Form sums of nonnegative self-adjoint operators. Acta Math. Hungar. 111, 81–105 (2006)
S. Hassi, A. Sandovici, H.S.V. de Snoo, H. Winkler, A general factorization approach to the extension theory of nonnegative operators and relations. J. Operator Theory 58, 351–386 (2007)
S. Hassi, A. Sandovici, H.S.V. de Snoo, H. Winkler, Extremal extensions for the sum of nonnegative self-adjoint relations. Proc. Am. Math. Soc. 135, 3193–3204 (2007)
S. Hassi, H.S.V. de Snoo, Factorization, majorization, and domination for linear relations. Ann. Univ. Sci. Budapest 58, 55–72 (2015)
S. Hassi, H.S.V. de Snoo, F.H. Szafraniec, Componentwise and canonical decompositions of linear relations. Dissertationes Mathematicae 465, 59 (2009)
T. Kato, Perturbation Theory for Linear Operators. Corrected Printing of the Second Edition (Springer, Berlin, 1980)
M. Roman, A. Sandovici, The square root of nonnegative self-adjoint linear relations in Hilbert spaces. J. Oper. Theory 82(2), 357–367 (2019)
A. Sandovici, On polynomials in a linear relation, Proceedings of the 4th Workshop on Operator Theory in Krein Spaces and Applications, Berlin, December 17-19, 2004. In volume Operator Theory in Inner Product Spaces, Series: OTAA, Vol. 175, Förster, K.-H.; Jonas, P.; Langer, H.; Trunk, C. (Eds.) 2007, VI, 240 p., ISBN: 978-3-7643-8269-8, 231–240
A. Sandovici, On domains and ranges of powers of linear relations in linear spaces. Complex Anal. Oper. Theory 6(3), 749–758 (2012)
A. Sandovici, Von Neumann’s theorem for linear relations. Linear Multilinear Algebra 66(9), 1750–1756 (2018)
A. Sandovici, Self-adjointness and skew-adjointness criteria involving powers of linear relations. J. Math. Anal. Appl. 470, 186–200 (2019)
K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics 265 (Springer, Dordrecht, 2012)
Z. Sebestyén, J. Stochel, Restrictions of positive self-adjoint operators. Acta Sci. Math. (Szeged) 55, 149–154 (1991)
Z. Sebestyén, Zs Tarcsay, \(T^{*}T\) always has a positive self-adjoint extension. Acta Math. Hungar. 135, 116–129 (2012)
Z. Sebestyén, Zs Tarcsay, Characterizations of self-adjoint operators. Stud. Sci. Math. Hungar. 50, 423–435 (2013)
Z. Sebestyén, Zs Tarcsay, Characterizations of essentially self-adjoint and skew-adjoint operators. Stud. Sci. Math. Hungar. 52(3), 371–385 (2015)
Z. Sebestyén, Zs Tarcsay, Adjoint of sums and products of operators in Hilbert spaces. Acta Sci. Math. (Szeged) 82(1–2), 175–191 (2016)
Z. Sebestyén, Zs Tarcsay, Operators having self-adjoint squares. Annales Univ. Sci. Budapest. Sect. Math. 58, 105–110 (2015)
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Roman, M., Sandovici, A. Essentially self-adjoint linear relations in Hilbert spaces. Period Math Hung 83, 122–132 (2021). https://doi.org/10.1007/s10998-020-00373-8
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DOI: https://doi.org/10.1007/s10998-020-00373-8
Keywords
- Hilbert space
- Nonnegative linear relation
- Symmetric linear relation
- Essentially self-adjoint linear relation