## On the regularity of the critical point infinity of definitizable operators.(English)Zbl 0572.47023

For a definitizable operator A in a Krein space $${\mathcal K}$$ [for the terminology and fundamental results on this topic see: H. Langer, Proc. Conf., Dubrovnik, Yugoslavia 1981, Lect. Notes Math. 948, 1-46 (1982; Zbl 0511.47023)] several criteria for the regularity of the critical point (c.p.) $$\infty$$ are given. In particular it is shown that $$\infty$$ is not a singular c.p. of A if and only if in the Krein space $${\mathcal K}$$ there exists a positive, bounded and boundedly invertible operator W such that W$${\mathcal D}(A)\subseteq {\mathcal D}(A)$$ (or W$${\mathcal D}[JA]\subseteq {\mathcal D}[JA]$$, where $${\mathcal D}[JA]={\mathcal D}(| JA|^{1/2})$$ and J is a fundamental symmetry on $${\mathcal K})$$. These criteria are used to prove that the regularity of the c.p. $$\infty$$ is preserved under some additive perturbations, as well as for certain operators which are related to A. Applications to indefinite Sturm- Liouville problems and connections with the results of R. Beals [J. Differ. Equations 56, 391-407 (1985; Zbl 0512.34017)] are indicated.

### MSC:

 47B50 Linear operators on spaces with an indefinite metric 34L99 Ordinary differential operators

### Citations:

Zbl 0511.47023; Zbl 0512.34017
Full Text:

### References:

 [1] Bayasgalan, Ts.: On the fundamental reducibility of positive operators in spaces with indefinite metric. (Russian.) Studia Sci. Math. Hungar. 13 (1978), 143–150. [2] Beals, R.: Indefinite Sturm-Liouville problems and half-range completeness. J. Differential Equations, to appear. · Zbl 0512.34017 [3] Bognár, J.: Indefinite inner product spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete 78, Springer-Verlag, Berlin/Heidelberg/New York, 1974. [4] Coddington, E.A., de Snoo, H.S.V.: Regular boundary value problems associated with pairs of ordinary differential expressions. Lecture Notes in Mathematics 858, Springer-Verlag, Berlin/Heidelberg/New York, 1981. · Zbl 0464.34003 [5] Ćurgus, B., Langer, H.: Spectral properties of selfadjoint ordinary differential operators with an indefinite weight function. Proceedings of the 1984 Workshop ”Spectral theory of Sturm-Liouville differential operators”. ANL-84-73, Argonne National Laboratory, Argonne, Illinois, 1984, 73–80. [6] Hess, P.: Zur Theorie der linearen Operatoren eines J-Raumes. Operatoren, die von kanonischen Zerlegungen reduziert werden. Math. Z. 106 (1968), 88–96. · Zbl 0165.47901 [7] Jonas, P.: Über die Erhaltung der Stabilität J-positiver Operatoren bei J-positiven und J-negativen Störungen. Math. Nachr. 65 (1975), 211–218. · Zbl 0328.47018 [8] Jonas, P.: Relatively compact perturbations of uniformly J-positive operators. Akademie der Wissenschaften der DDR, Zentralinstitut für Mathematik und Mechanik, Preprint P-15/80, Berlin, 1980. · Zbl 0444.47029 [9] Jonas, P.: On the functional calculus and the spectral function for definitizable operators in Krein spaces. Beiträge Anal. 16 (1981), 121–135. · Zbl 0556.47019 [10] Jonas, P.: Compact perturbations of definitizable operators. II. J. Operator Theory 8 (1982), 3–18. · Zbl 0565.47021 [11] Jonas, P.: Regularity criteria for critical points of definitizable operators. Spectral theory of linear operators and related topics, 8th international conference on operator theory, Timisoara and Herculane (Romania), June 6–16, 1983. Operator Theory: Advances and Applications 14, Birkhäuser Verlag, Basel/Boston/Stuttgart, 1984, 179–195. [12] Kaper, H.G., Kwong, M.K., Lekkerkerker, C.G., Zettl, A.: Full- and partial-range eigenfunction expansions for Sturm-Liouville problems with indefinite weights. Proc. Roy. Soc. Edinburgh Sect. A 98 (1984), 69–88. · Zbl 0558.47019 [13] Kato, T.: Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 132, Springer-Verlag, Berlin/Heidelberg/New York, 1966. · Zbl 0148.12601 [14] Kreîn, M.G.: The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. I–II. (Russian.) Mat. Sb. (N.S.) 20 (62) (1947), 431–495, and 21 (63) (1947), 365–404. · Zbl 0029.14103 [15] Kreîn, M.G.: Completely continuous linear operators in function spaces with two norms. (Ukrainian.) Akad. Nauk Ukrain. RSR. Zbirnik Prac’ Inst. Mat. 9 (1947), 104–129. [16] Kreîn, S.G.: Linear differential equations in Banach space. (Russian.) Izdatel’stvo ”Nauka”, Moscow/Leningrad, 1967. [17] Langer, H.: Maximal dual pairs of invariant subspaces of J-selfadjoint operators. (Russian.) Mat. Zametki 7 (1970), 443–447. · Zbl 0192.47702 [18] Langer, H.: Verallgemeinerte Resolventen eines J-nichtnegativen Operators mit endlichem Defekt. J. Funct. Anal. 8 (1971), 287–320. · Zbl 0232.47008 [19] Langer, H.: Spectral functions of definitizable operators in Krein spaces. Functional analysis, Proceedings of a conference held at Dubrovnik, Yugoslavia, November 2–14, 1981. Lecture Notes in Mathematics 948, Springer-Verlag, Berlin/Heidelberg/New York, 1982, 1–46. [20] Lax, P.D.: Symmetrizable linear transformations. Comm. Pure Appl. Math. 7 (1954), 633–647. · Zbl 0057.34402 [21] Reid, W.T.: Symmetrizable completely continuous linear transformations in Hilbert space. Duke Math. J. 18 (1951), 41–56. · Zbl 0042.36002 [22] Veselić, K.: On spectral properties of a class of J-selfadjoint operators. I. Glasnik Mat. Ser. III 7 (1972), 229–248. · Zbl 0249.47027 [23] Weidmann, J.: Linear operators in Hilbert spaces. Graduate Texts of Mathematics 68, Springer-Verlag, New York/Heidelberg/Berlin, 1980. · Zbl 0434.47001
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