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Arlinskii's Iteration and its Applications

Part of: General theory of linear operators

Published online by Cambridge University Press:  29 August 2018

Tamás Titkos
Tamás Titkos*
Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Reáltanoda u. 13–15, Hungary ([email protected])
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Abstract

Several Lebesgue-type decomposition theorems in analysis have a strong relation to the operation called the parallel sum. The aim of this paper is to investigate this relation from a new point of view. Namely, using a natural generalization of Arlinskii's approach (which identifies the singular part as a fixed point of a single-variable map) we prove the existence of a Lebesgue-type decomposition for non-negative sesquilinear forms. As applications, we also show how this approach can be used to derive analogous results for representable functionals, non-negative finitely additive measures, and positive definite operator functions. The focus is on the fact that each theorem can be proved with the same completely elementary method.

Keywords

parallel sumLebesgue decompositionsingularity

MSC classification

Primary: 47A07: Forms (bilinear, sesquilinear, multilinear)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 1 , February 2019 , pp. 125 - 133
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1.Anderson, W. N. Jr. and Duffin, R. J., Series and parallel addition of matrices, J. Math. Anal. Appl. 26 (1969), 576–594.Google Scholar
2.Ando, T., Lebesgue-type decomposition of positive operators, Acta. Sci. Math. (Szeged) 38 (1976), 253–260.Google Scholar
3.Ando, T. and Szymański, W., Order structure and Lebesgue decomposition of positive definite operator functions, Indiana Univ. Math. J. 35 (1986), 157–173.Google Scholar
4.Arendt, W. and ter Elst, A. F. M., Sectorial forms and degenerate differential operators, J. Operator Theory 67 (2012), 33–72.Google Scholar
5.Arlinskii, Yu. M., On the mapping connected with parallel addition of nonnegative operators, Positivity 21 (2017), 299–327.Google Scholar
6.Bot, R. I. and László, Sz., On the generalized parallel sum of two maximal monotone operators of Gossez type (D), J. Math. Anal. Appl. 391 (2012), 82–98.Google Scholar
7.Darst, R. B., A decomposition of finitely additive set functions, J. Reine Angew. Math. 210 (1962), 31–37.Google Scholar
8.Di Bella, S. and Trapani, C., Some representation theorems for sesquilinear forms, J. Math. Anal. Appl. 451 (2017), 64–83.Google Scholar
9.Djikić, M. S., Extensions of the Fill–Fishkind formula and the infimum parallel sum relation, Linear Multilinear Algebra 64 (2016), 2335–2349.Google Scholar
10.Djikić, M. S. and Djordjević, D. S., Coherent and precoherent elements in Rickart *-rings, Linear Algebra Appl. 509 (2016), 64–81.Google Scholar
11.Eriksson, S. L. and Leutwiler, H., A potential theoretic approach to parallel addition, Math. Ann. 274 (1986), 301–317.Google Scholar
12.Gheondea, A., Gudder, S. and Jonas, P., On the infimum of quantum effects, J. Math. Phys. 46 (2005), 062102.Google Scholar
13.Gudder, S., A Radon-Nikodym theorem for *-algebras, Pacific J. Math. 80 (1979), 141–149.Google Scholar
14.Hassi, S., Sebestyén, Z. and de Snoo, H., Lebesgue type decompositions for nonnegative forms, J. Funct. Anal. 257 (2009), 3858–3894.Google Scholar
15.Kosaki, H., Lebesgue decomposition of states on a von Neumann algebra, Am. J. Math. 107 (1985), 697–735.Google Scholar
16.Palmer, T. W., Banach algebras and the general theory of *-Algebras II (Cambridge University Press, 2001).Google Scholar
17.Passty, G. B., The parallel sum of nonlinear monotone operators, Nonlinear Anal. 10 (1986), 215–227.Google Scholar
18Pekarev, È. L. and Šmulj́an, Yu. L., Parallel addition and parallel subtraction of operators, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), 366–387.Google Scholar
19.Rao, K. P. S. B. and Rao, M. B., Theory of charges (Academic Press, 1983).Google Scholar
20.Sebestyén, Z., Tarcsay, Zs. and Titkos, T., Lebesgue decomposition theorems, Acta Sci. Math. (Szeged) 79 (2013), 219–233.Google Scholar
21.Simon, B., A canonical decomposition for quadratic forms with applications to monotone convergence theorems, J. Funct. Anal. 28 (1978), 377–385.Google Scholar
22.Szymański, W., Positive forms and dilations, Trans. Amer. Math. Soc. 301 (1987), 761–780.Google Scholar
23.Tarcsay, Zs., On the parallel sum of positive operators forms, and functionals, Acta Math. Hungar. 147 (2015), 408–426.Google Scholar
24.Tarcsay, Zs., Lebesgue decomposition for representable functionals on *-algebras, Glasgow Math. J. 58 (2016), 491–501.Google Scholar
25.ter Elst, A. F. M. and Sauter, M. J., The regular part of second-order differential sectorial forms with lower-order terms, J. Evol. Equ. 13 (2013), 737–749.Google Scholar
26.Vogt, H., The regular part of symmetric forms associated with second-order elliptic differential expressions, Bull. Lond. Math. Soc. 41 (2009), 441–444.Google Scholar
27.von Neumann, J., Zur Theorie der Unbeschränkten Matrizen, J. Reine Angew. Math. 161 (1929), 208–236.Google Scholar