Abstract
The adjoint of a linear operator or relation from a Hilbert space \( \mathfrak{H} \) to a Hilbert space \( \mathfrak{K} \) is a closed linear relation. The domain and the range of the adjoint are characterized in terms of certain mappings defined on K and \( \mathfrak{H} \), respectively. These characterizations are applied to contractions between Hilbert spaces and to the form domains and ranges of the Friedrichs and Kreĭn-von Neumann extensions of a nonnegative operator or relation. Furthermore these characterizations are used to introduce and derive properties of the parallel sum and the parallel difference of a pair of forms on a linear space.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
W.N. Anderson, Jr., “Shorted operators”, SIAM J. Appl. Math., 20 (1971), 520–525.
W.N. Anderson and R.J. Duffin, “Series and parallel addition of matrices”, J. Math. Anal. Appl., 26 (1969), 576–594.
W.N. Anderson and G.E. Trapp, “Shorted operators. II”, SIAM J. Appl. Math., 28 (1975), 60–71.
T. Ando, “Lebesgue-type decomposition of positive operators”, Acta Sci. Math. (Szeged), 38 (1976), 253–260.
T. Ando and K. Nishio, “Positive selfadjoint extensions of positive symmetric operators”, Tôhoku Math. J., 22 (1970), 65–75.
Yu.M. Arlinskiĭ, S. Hassi, Z. Sebestyén, and H.S.V. de Snoo, “On the class of extremal extensions of a nonnegative operator”, Oper. Theory Adv. Appl. (B. Sz.-Nagy memorial volume), 127 (2001), 41–81.
L. de Branges and J. Rovnyak, Square summable power series, Holt, Rinehart and Winston, New York, 1966.
S.L. Eriksson and H. Leutwiler, “A potential theoretic approach to parallel addition”, Math. Ann., 274 (1986), 301–317.
P. Fillmore and J. Williams, “On operator ranges”, Adv. in Math., 7 (1971), 254–281.
S. Hassi, “On the Friedrichs and the Kreĭn-von Neumann extension of nonnegative relations”, Acta Wasaensia, 122 (2004), 37–54.
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, “A general factorization approach to the extension theory of nonnegative operators and relations”, J. Operator Theory, 58 (2007), 351–386.
S. Hassi, Z. Sebestyén, and H.S.V. de Snoo, “Lebesgue type decompositions for nonnegative forms”, to appear in J. Functional Analysis.
E. Jaimes, “On ranges of adjoint operators”, Ann. Univ. Sci. Budapest, 33 (1990), 235–237.
E.L. Pekarev, “Shorts of operators and some extremal problems”, Acta Sci. Math. (Szeged), 56 (1992), 147–163.
E.L. Pekarev and Yu.L. Shmulyan, “Parallel addition and parallel subtraction of operators”, Izv. Akad. Nauk SSSR, Ser Mat., 40 (1976), 366–387.
Z. Sebestyén, “On ranges of adjoint operators in Hilbert space”, Acta Sci. Math. (Szeged), 46 (1983), 295–298.
Yu.L. Shmulyan, “Two-sided division in a ring of operators”, Math. Notes 1 (1967), 400–403.
B. Simon, “A canonical decomposition for quadratic forms with applications to monotone convergence theorems”, J. Functional Analysis, 28 (1978), 377–385.
B. Sz.-Nagy and C. Foia§, Harmonic analysis of operators on Hilbert space, North-Holland, New York, 1970.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
To the memory of Peter Jonas
Rights and permissions
Copyright information
© 2009 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Hassi, S., Sebestyén, Z., de Snoo, H. (2009). Domain and Range Descriptions for Adjoint Relations, and Parallel Sums and Differences of Forms. In: Behrndt, J., Förster, KH., Trunk, C. (eds) Recent Advances in Operator Theory in Hilbert and Krein Spaces. Operator Theory: Advances and Applications, vol 198. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0180-1_11
Download citation
DOI: https://doi.org/10.1007/978-3-0346-0180-1_11
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0346-0179-5
Online ISBN: 978-3-0346-0180-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)