Abstract
A linear relation, i. e., a multivalued operator T from a Hilbert space H to a Hilbert space K has Lebesgue type decompositions T T1 is a closable operator and T2 is an operator or relation which is singular. There is one canonical decomposition, called the Lebesgue decomposition of T, whose closable part is characterized by its maximality among all closable parts in the sense of domination. All Lebesgue type decompositions are parametrized, which also leads to necessary and sufficient conditions for the uniqueness of such decompositions. Similar results are given for weak Lebesgue type decompositions, where T1 is just an operator without being necessarily closable. Moreover, closability is characterized in different useful ways. In the special case of range space relations the above decompositions may be applied when dealing with pairs of (nonnegative) bounded operators and nonnegative forms as well as in the classical framework of positive measures.
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References
N. I. Achieser and I.M. Glasman, Theorie der linearen Operatoren im Hilbertraum, 8th edition, Akademie Verlag, Berlin, 1981.
T. Ando, Lebesgue-type decomposition of positive operators, Acta Sci. Math. (Szeged), 38 (1976), 253–260.
R. Arens, Operational calculus of linear relations, Pacific J. Math., 11 (1961), 9–23.
Yu. Arlinski˘ı, On the mappings connected with parallel addition of nonnegative operators, Positivity, 21 (2017), 299–327.
J. Behrndt, S. Hassi, H.S.V. de Snoo and R. Wietsma, Monotone convergence theorems for semibounded operators and forms with applications, Proc. Royal Soc. Edinburgh, 140A (2010), 927–951.
P. Fillmore and J. Williams, On operator ranges, Adv. Math., 7 (1971), 254–281.
I. Gohberg and M. G. Kre˘ın, The basic propositions on defect numbers, root numbers and indices of linear operators, Uspekhi Mat. Nauk., 12 (1957), 43–118 (Russian) [English translation: Transl. Amer. Math. Soc. (2), 13 (1960), 185–264].
S. Hassi, A. Sandovici, H.S.V. de Snoo and H. Winkler, Form sums of nonnegative selfadjoint operators, Acta Math. Hungar., 111 (2006), 81–105.
S. Hassi, Z. Sebestyén and H.S.V. de Snoo, Domain and range descriptions for adjoint relations, and parallel sums and differences of forms, Oper. Theory Adv. Appl., 198 (2009), 211–227.
S. Hassi, Z. Sebestyén and H.S.V. de Snoo, Lebesgue type decompositions for nonnegative forms, J. Functional Analysis, 257 (2009), 3858–3894.
S. Hassi, Z. Sebestyén, H.S.V. de Snoo and F.H. Szafraniec, A canonical decomposition for linear operators and linear relations, Acta Math. Hungarica, 115 (2007), 281–307.
S. Hassi and H.S.V. de Snoo, Factorization, majorization, and domination for linear relations, Annales Univ. Sci. Budapest, 58 (2015), 53–70.
S. Hassi and H.S.V. de Snoo, Lebesgue type decompositions and Radon-Nikodym derivatives for pairs of bounded linear operators, in preparation.
S. Hassi, H.S.V. de Snoo and F.H. Szafraniec, Componentwise and Cartesian decompositions of linear relations, Dissertationes Mathematicae, 465 (2009), 59 pages.
S. Izumino, Decomposition of quotients of bounded operators with respect to closability and Lebesgue-type decomposition of positive operators, Hokkaido Math. J., 18 (1989), 199–209.
P.E.T. Jorgensen, Unbounded operators; perturbations and commutativity problems, J. Functional Analysis, 39 (1980), 281–307.
T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin, 1980.
S. Ôta, Decomposition of unbounded derivations in operator algebras, Tôhoku Math. J., 33 (1981), 215–225.
S. Ôta, Closed linear operators with domain containing their range, Proc. Edinburgh Math. Soc., 27 (1984), 229–233.
S. Ôta, On a singular part of an unbounded operator, Zeitschrift für Analysis und ihre Anwendungen, 7 (1987), 15–18.
D. Popovici and Z. Sebestyén, Factorizations of linear relations, Adv. Math., 233 (2013), 40–55.
Z. Sebestyén, Zs. Tarcsay and T. Titkos, Lebesgue decomposition theorems, Acta Sci. Math. (Szeged), 79 (2013), 219–233.
Z. Sebestyén, Zs. Tarcsay and T. Titkos, A short-type decomposition of forms, Operators Matrices, 9 (2015), 815–830.
B. Simon, A canonical decomposition for quadratic forms with applications to monotone convergence theorems, J. Functional Analysis, 28 (1978), 377–385.
M.H. Stone, Linear transformations in Hilbert space and their applications to analysis, Amer. Math. Soc. Colloquium Publications, Vol. 15, Amer. Math. Soc., 1932.
Zs. Tarcsay, Lebesgue-type decomposition of positive operators, Positivity, 17 (2013), 803–817.
Zs. Tarcsay, Radon–Nikodym theorems for nonnegative forms, measures and representable functionals, Complex Anal. Oper. Theory, 10 (2016), 479–494.
T. Titkos, Arlinskii’s iteration and its applications, Proc. Edinburgh Math. Soc., pp. 9; doi: 10. 1017/S0013091518000287.
J. Weidmann, Lineare Operatoren im Hilberträumen, B.G. Teubner, Stuttgart, 1976.
D. Werner, Funktionalanalysis, 3. Auflage, Springer, 2000.
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Communicated by L. Molnár
To Professor Ando on the occasion of his 85th birthday
The research was partially supported by a grant from the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters.
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Hassi, S., Sebestyén, Z. & de Snoo, H. Lebesgue type decompositions for linear relations and Ando’s uniqueness criterion. ActaSci.Math. 84, 465–507 (2018). https://doi.org/10.14232/actasm-018-757-0
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DOI: https://doi.org/10.14232/actasm-018-757-0