S ?/sub> relative to some boundary triples \({\big(\mathbb{C},\,\Gamma_0^+,\,\Gamma_1^+\big)}\) and \({\big(\mathbb{C},\,\Gamma_0^-,\,\Gamma_1^-\big)}\). Sufficient conditions for a nonnegative partially fundamentally reducible operator A to be similar to a self-adjoint operator in a Hilbert space are given in terms of the Weyl functions m + and m ?/sub> of S + and S ?/sub> relative to the boundary triples \({\big(\mathbb{C},\,\Gamma_0^+,\,\Gamma_1^+\big)}\) and \({\big(\mathbb{C},\,\Gamma_0^-,\Gamma_1^-\big)}\). Moreover, it is shown that under some asymptotic assumptions on m + and m ?/sub> all positive self-adjoint extensions of the operator S are similar to self-adjoint operators in a Hilbert space." />
Partially Fundamentally Reducible Operators in Kre?n Spaces
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  • 作者：Branko ?urgus ; Vladimir Derkach
  • 关键词：Primary 47B50 ; Secondary 46C20 ; 47B25 ; Self ; adjoint extension ; symmetric operator ; Kre?n space ; fundamentally reducible operator ; coupling of operators ; boundary triple ; Weyl function ; similar to a self ; adjoint operator
  • 刊名：Integral Equations and Operator Theory
  • 出版年：2015
  • 出版时间：July 2015
  • 年：2015
  • 卷：82
  • 期：4
  • 页码：469-518
  • 全文大小：984 KB
  • 参考文献：1.Akhiezer N.I., Glazman I.M.: Theory of Linear Operators in Hilbert Space, Two Volumes Bound as One. Dover Publications, New York (1993)
    2.Akhiezer N.I., Glazman I.M.: Theory of Linear Operators in Hilbert Space. (Russian) vol. II, Third edition. Vishcha Shkola, Kharkov (1978)
    3.Alpay D., Gohberg I.: Pairs of selfadjoint operators and their invariants. St. Petersburg Math. J. 16, 59-04 (2005)MathSciNet View Article
    4.Azizov T.Y., Behrndt J., Trunk C.: On finite rank perturbations of definitizable operators. J. Math. Anal. Appl. 339, 1161-168 (2008)MathSciNet View Article
    5.Azizov T.Y., Iokhvidov I.S.: Linear Operators in Spaces with an Indefinite Metric. John Wiley & Sons, London (1990)
    6.Bayasgalan, T.: On the fundamental reducibility of positive operators in spaces with indefinite metric. (Russian) Studia Sci. Math. Hungar. 13 (1978), 143-50 (1981)
    7.Behrndt J.: On the spectral theory of singular indefinite Sturm-Liouville operators. J. Math. Anal. Appl. 334, 1439-449 (2007)MathSciNet View Article
    8.Behrndt J., Hassi S., de Snoo H., Wietsma R., Winkler H.: Linear fractional transformations of Nevanlinna functions associated with a nonnegative operator. Complex Anal. Oper. Theory 7, 331-62 (2013)MathSciNet View Article
    9.Behrndt J., Philipp F.: Spectral analysis of singular ordinary differential operators with indefinite weights. J. Differ. Equ. 248, 2015-037 (2010)MathSciNet View Article
    10.Bennewitz C.: Spectral asymptotics for Sturm-Liouville equations. Proc. London Math. Soc. 59, 294-38 (1989)MathSciNet View Article
    11.Bognar J.: Indefinite Inner Product Spaces. Springer, New York (1974)View Article
    12.Carmichael R.D.: Asymptotic analysis for complex-valued Stieltjes transforms. Integr. Transforms Spec. Funct. 22, 277-82 (2011)View Article
    13.Carmichael R.D., Hayashi E.K.: Abelian theorems for the Stieltjes transform of functions. II. Internat. J. Math. Math. Sci. 4, 67-8 (1981)MathSciNet View Article
    14.Coddington E.A., Levinson N.: Theory of Ordinary Differential Equations. McGraw-Hill Book Company, New York (1955)
    15.?urgus B.: On the regularity of the critical point infinity of definitizable operators. Integr. Equ Oper. Theory 8, 462-88 (1985)View Article
    16.?urgus B., Dijksma A., Read T.: The linearization of boundary eigenvalue problems and reproducing kernel Hilbert spaces. Linear Algebra Appl. 329, 97-36 (2001)MathSciNet View Article
    17.?urgus B., Fleige A., Kostenko A.: The Riesz basis property of an indefinite Sturm-Liouville problem with non-separated boundary conditions. Integr. Equ. Oper. Theory 77, 533-57 (2013)View Article
    18.?urgus B., Langer H.: A Kre?n space approach to symmetric ordinary differential operators with an indefinite weight function. J. Differ. Equ. 79, 31-1 (1989)View Article
    19.?urgus B., Najman B.: The operator \({({\rm sgn}\, x){d^{2}}/{dx^{2}}}\) is similar to a self-adjoint operator in \({L^{2}({\mathbb{R}})}\) . Proc. Am. Math. Soc. 123, 1125-128 (1995)
    20.Derkach V.A., Malamud M.M.: Generalized resolvents and the boundary value problems for hermitian operators with gaps, J. Funct. Anal. 95, 1-5 (1991)MathSciNet View Article
    21.Derkach V.A., Malamud M.M.: The extension theory of Hermitian operators and the moment problem. Anal. J. Math. Sci. 73, 141-42 (1995)MathSciNet View Article
    22.Derkach V.A.: On Weyl function and generalized resolvents of a Hermitian operator in a Kre?n space. Integr. Equ. Oper. Theory 23, 387-15 (1995)MathSciNet View Article
    23.Derkach V.A., Hassi S., Malamud M.M., de Snoo H.S.V.: Generalized resolvents of symmetric operators and admissibility. Methods Funct. Anal. Topology 6, 24-5 (2000)MathSciNet
    24.Donoghue W.F.: Monotone Matrix Functions and Analytic Continuation Die Grundlehren der mathematischen Wissenschaften, Band 207. Springer-Verlag, Berlin (1974)View Article
    25.Everitt W.N.: On a property of the m-coefficient of a second-order linear differential equation. J. London Math. Soc. 4(2), 443-57 (1971/1972)
    26.Everitt W.N., Zettl A.: On a class of integral inequalities. J. London Math. Soc. 17(2), 291-03 (1978/1979)
    27.Faddeev, M.M., Shterenberg, R.G.: On the similarity of some singular differential operators to selfadjoint operators. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 270 (2000, Russian), Issled. po Linein. Oper. i Teor. Funkts. 28, 336-49, 370-71; translation in J. Math. Sci. (N. Y.) 115 (2003) 2279-286
    28.Fleige A.: A counterexample to completeness properties for indefinite Sturm-Liouville problems. Math. Nachr. 190, 123-28 (1998)MathSciNet View Article
    29.Fleige A., Najman B.: Nosingularity of Critical Points of Some Differential and Difference Operators, Oper. Theory: Adv. Appl., vol. 102. Birkh?user, Basel (1998)
    30.Garling D.J.H.: Inequalities: A Journey into Linear Analysis. Cambridge University Press, Cambridge (2007)View Article
    31.Glazman, I.M.: Direct methods of
  • 作者单位：Branko ?urgus (1)
    Vladimir Derkach (2) (3)

    1. Department of Mathematics, Western Washington University, 516 High Street, Bellingham, WA, 98226, USA
    2. Department of Mathematics, Donetsk National University, 600-Richchya Str 21, Vinnytsya, 21021, Ukraine
    3. Department of Mathematics, National Pedagogical University, Pirogova Str 9, Kiev, 01601, Ukraine
  • 刊物类别：Mathematics and Statistics
  • 刊物主题：Mathematics
    Analysis
  • 出版者：Birkh盲user Basel
  • ISSN：1420-8989
文摘
A self-adjoint operator A in a Kre?n space (\({\mathcal{K}, [\cdot , \cdot]}\)) is called partially fundamentally reducible if there exist a fundamental decomposition \({\mathcal{K} = \mathcal{K}_{+}[\dot{+}]\mathcal{K}_{-}}\) (which does not reduce A) and densely defined symmetric operators S + and S ?/sub> in the Hilbert spaces (\({\mathcal{K}_+, [\cdot , \cdot]}\)) and \({(\mathcal{K}_-, -[\cdot , \cdot])}\), respectively, such that each S + and S ?/sub> has defect numbers (1, 1) and the operator A is a self-adjoint extension of \({S = S_{+} \oplus (-S_-)}\) in the Kre?n space \({(\mathcal{K}, [\cdot , \cdot])}\). The operator A is interpreted as a coupling of operators S + and ?em class="EmphasisTypeItalic">S ?/sub> relative to some boundary triples \({\big(\mathbb{C},\,\Gamma_0^+,\,\Gamma_1^+\big)}\) and \({\big(\mathbb{C},\,\Gamma_0^-,\,\Gamma_1^-\big)}\). Sufficient conditions for a nonnegative partially fundamentally reducible operator A to be similar to a self-adjoint operator in a Hilbert space are given in terms of the Weyl functions m + and m ?/sub> of S + and S ?/sub> relative to the boundary triples \({\big(\mathbb{C},\,\Gamma_0^+,\,\Gamma_1^+\big)}\) and \({\big(\mathbb{C},\,\Gamma_0^-,\Gamma_1^-\big)}\). Moreover, it is shown that under some asymptotic assumptions on m + and m ?/sub> all positive self-adjoint extensions of the operator S are similar to self-adjoint operators in a Hilbert space.

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