×

Generalized resolvents and the boundary value problems for Hermitian operators with gaps. (English) Zbl 0748.47004

Let \(A\) be a Hermitian operator with gaps \((\alpha_ j,\beta_ j)\), \(j=1,2,\dots,m\). The main result of the paper (Section 8) is the description of all self-adjoint extensions of \(A\) putting exactly \(k_ j<+\infty\) eigenvalues into the gap \((\alpha_ j,\beta_ j)\), \(j=1,2,\dots,m\). In particular, for \(k_ j=0\), \(1,2,\dots,m\), this answered a question which was posed by M. G. Krein in 1945. The solution of the problem is given in terms of the so-called Weyl function which is an important object in the extension theory of Hermitian operators in the framework of abstract boundary conditions.
In Section 1 the Weyl function is introduced and its relation to the \(\mathcal D\)-function of M. G. Krein and the characteristic function of a Hermitian operator is clarified. Section 2 deals with the famous Krein resolvent formula for self-adjoint extensions of Hermitian operators. In Section 3 the extensions of a nonnegative operator \(A\geq 0\) are studied in detail. In particular, in terms of the Weyl function a characterization of the so-called Friedrichs and Krein extensions are given. In Section 4 the special case of one gap \((m=1)\) is investigated, i.e., extensions are studied which put exactly \(k\) eigenvalues into this gap. In order to prepare the main result of Section 8 in Section 5 new classes \(S^{\pm k}_{\mathcal H}\) of analytic functions are introduced which generalize the well-known Krein-Stieltjes classes \((S)_{\mathcal H}\) and \((S)^{-1}_{\mathcal H}\). Section 6 and 7 are devoted to the problem of generalized resolvents of Hermitian operators preserving the gap \((\alpha,\beta)\). We recall that the generalized resolvent is the compression of the resolvent of a self-adjoint extension of \(A\) in larger Hilbert space to the Hilbert space on which \(A\) acts. In Section 9 the results are used to study spectral properties of boundary value problems for some differential operators. At the end (Section 10) an application of the main theorems of Section 8 is given to the so-called Hamburger moment problem. In particular, a solvability criterion and a description of all solutions of this problem with the property that the supports are situated in \(\mathbb{R}^ 1\setminus\bigcup^ m_{j=1}(\alpha_ j,\beta_ j)\) are obtained in terms of the Nevanlinna matrix.
Reviewer: H.Neidhardt

MSC:

47A20 Dilations, extensions, compressions of linear operators
47A10 Spectrum, resolvent
47A57 Linear operator methods in interpolation, moment and extension problems
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Arov, D. Z.; Grossman, L. Z., Scattering matrices in the theory of extensions of isometric operators, Dokl. Acad. Nauk USSR, 270, No. 1, 17-20 (1983), [Russian] · Zbl 0543.47010
[2] Ahiezer, N. I., Classical Moment Problem (1961), Fizmatgiz: Fizmatgiz Moscow, [Russian]
[3] Ahiezer, N. I.; Glazman, I. M., Theory of Linear Operators on Hilbert Space (1966), Nauka: Nauka Moscow, [Russian]
[4] Berezanskii, Yu. M., Decomposition on eigenfunctions of selfadjoint operators (1965), Nauk. Dumka: Nauk. Dumka Kiev, [Russian] · Zbl 0142.37203
[5] Birman, M. S., On the selfadjoint extensions of positive definite operators, Mat. Sb., 38, 80, 431-450 (1956)
[6] Bruk, V. M., Math. USSR Sb., 29 (1976), Engl. transl. in · Zbl 0334.47010
[7] Bruk, V. M., On the extensions of a symmetric relation, Mat. Zametki, 22, No. 6, 825-834 (1977), [Russian] · Zbl 0379.47010
[8] Vishik, M. I., On general boundary problems for elliptic differential equations, (Trudy Moskov. Mat. Obshch., 1 (1952)), 187-246, [Russian]
[9] Gehtman, M. M., On the spectrum of selfadjoint extensions of a semibounded operator, Dokl. Acad. Nauk SSSR, 186, No. 6, 1250-1252 (1969), [Russian]
[10] Gehtman, M. M., On the existence of the superficial states for the extensions of a Laplace operator, Funktsional. Anal. i Prilozen., 16, No. 1, 62-63 (1982), [Russian]
[11] Gorbachuk, V. I.; Gorbachuk, M. L., Boundary Value Problems for Operator-Differential Equations (1984), Maukova Dumka: Maukova Dumka Kiev, [Russian] · Zbl 0567.47041
[12] Gorbachuk, M. L.; Kutovoi, V. A., On resolvent comparativeness of certain boundary problems, Funktional. Anal. i Prilozhen., 16, No. 3, 52-53 (1982), [Russian]
[13] Gorbachuk, M. L.; Mihailets, V. A., Semibounded selfadjoint extensions of symmetric operators, Dokl. Acad. Nauk SSSR, 226, No. 4, 765-767 (1976), [Russian]
[14] Gohberg, I. Ts.; Krein, M. G., Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space (1965), Nauka: Nauka Moscow, [Russian]
[15] Gohberg, I. Ts.; Sigal, E. I., Operator generalization of the theorem of logarithmical resude, Mat. Sb., 84, 607-630 (1971), [Russian]
[16] Dezin, A. A., General Questions of the Theory of Boundary Value Problems (1980), Nauka: Nauka Moscow, [Russian] · Zbl 0494.35084
[17] Derkach, V. A.; Malamud, M. M., Weyl Function of Hermitian Operator and Its Connection with Characteristic Function, (Preprint 85-9 (1985), Donetsk. Fiz.- Tekhn. Inst. Acad. Nauk Ukrain. SSR: Donetsk. Fiz.- Tekhn. Inst. Acad. Nauk Ukrain. SSR Donetsk), [Russian] · Zbl 0738.47020
[18] Derkach, V. A.; Malamud, M. M., Soviet Math. Dokl., 35, No. 2 (1987), Engl. transl. in · Zbl 0655.47005
[19] Derkach, V. A.; Malamud, M. M., On one generalization of Krein-Stieltjes class, Manuscript No. 3309 (1987), [Russian] · Zbl 0655.47005
[20] Derkach, V. A.; Malamud, M. M., Weyl function of the Hermitian operator: Boundary-value problems for the semibounded operators, Manuscript No. 779 (1988), [Russian] · Zbl 0698.47004
[21] Derkach, V. A.; Malamud, M. M., Generalized resolvents of Hermitian operator with gaps, Manuscript No. 2238 (1988), [Russian] · Zbl 0748.47004
[22] Derkach, V. A.; Malamud, M. M., On certain classes of solutions of moment problem, Manuscript No. 2239 (1988), [Russian] · Zbl 0698.47004
[23] Derkach, V. A.; Malamud, M. M., On certain classes of analytic operator-valued functions with non-negative imaginary part, Dokl. Acad. Nauk Ukrain. SSR. Ser. A, No. 3, 13-17 (1989) · Zbl 0692.47016
[24] Derkach, V. A.; Malamud, M. M., Generalized resolvents and boundary-value problems for Hermitian operator with gaps, (Preprint 88.59 (1988), Inst. Matem. Akad. Nauk USSR: Inst. Matem. Akad. Nauk USSR Kiev), [Russian] · Zbl 0748.47004
[25] Kochubei, A. N., Math. Notes, 17 (1975), Engl. transl.
[26] Kochubei, A. N., Amer. Math. Soc. Transl., 124, 139-142 (1984), Engl. transl. in · Zbl 0555.47029
[27] Kochubei, A. N., Soviet J. Contemporary Math. Anal., 15 (1980), Engl. transl. in
[28] Kochubei, A. N., On extensions of j-symmetric operators, Tea. Funktsiǐ Funktsional. Anal. i Prilozhen, 31, 74-80 (1979), [Russian]
[29] Krein, M. G., On Hermitian operators with defect numbers one, Dokl. Akad. Nauk SSSR, 43, No. 8, 339-342 (1944), [Russian]
[30] Krein, M. G., On resolvent of Hermitian operator with defect numbers (m, m), Dokl. Akad. Nauk SSSR, 52, No. 8, 657-660 (1946)
[31] Krein, M. G., Theory of selfadjoint extensions of semibounded operators, Mat. Sb., 20, 431-498 (1947), [Russian]
[32] Krein, M. G., On one generalization of investigations of Stieltjes, Dokl. Akad. Nauk SSSR, 86, No. 6, 881-884 (1952), [Russian] · Zbl 0049.34702
[33] Krein, M. G.; Krasnoselskii, M. A., General theorems about extensions of Hermitian operators, Uspekhi Mat. Nauk, 2, No. 3, 60-106 (1947), [Russian]
[34] Krein, M. G.; Langer, G. K., Functional Anal. Appl., 5 (1971), Engl. transl. in · Zbl 0236.47034
[35] Krein, M. G.; Nudel’man, A. A., The Markov Moment Problem and Extremal Problems (1977), Amer. Math. Soc: Amer. Math. Soc Providence, RI, Moscow, 1973; Engl. transl. · Zbl 0361.42014
[36] Krein, M. G.; Obcharenko, I. E., Soviert Math. Dokl., 17 (1976), Engl. transl. in · Zbl 0362.47009
[37] Krein, M. G.; Ovcharenko, I. E., Siberian Math. J., 18 (1977), Engl. transl. in · Zbl 0384.47006
[38] Krein, M. G.; Ovcharenko, I. E., Soviet Math, Dokl., 18, No. 5 (1978), Engl. transl. in · Zbl 0409.47013
[39] Krein, M. G.; Saakian, Sh., Soviet Math. Dokl., 7 (1966), Engl. transl. in
[40] Krein, M. G.; Saakian, Sh. N., Functional Anal. Appl., 4 (1970), Engl. transl. in
[41] Krein, S. G.; Trofimov, V. P., On multyplicity of characteristic point of holomorphic operator-valued function, Mat. Issled., 5, No. 2, 105-114 (1970), [Russian]
[42] Levitan, B. M., The Inverse Sturm-Liouville Problems (1984), Nauka: Nauka Moscow, [Russian] · Zbl 0575.34001
[43] Livshits, M. S., On one class of linear operators in Hilbert space, Mat. Sb., 19, No. 2, 239-260 (1946), [Russian] · Zbl 0061.25903
[44] Marchenko, V. A., Sturm-Liouville Operators and Applications (1977), Naukova Dumka: Naukova Dumka Kiev, [Russian] · Zbl 0399.34022
[45] Mihailets, V. A., Spectrums of operators and boundary-value problems, (Spectral Analysis of Differential Operators (1980), Naukova Dumka: Naukova Dumka Kiev), 106-131, [Russian]
[46] Najmark, M. A., On spectral functions of a symmetric operator, Izv. Akad. Nauk SSSR Ser. Mat., 7, 285-296 (1943), [Russian] · Zbl 0061.26005
[47] Naimark, M. A., Linear Differential Operators (1969), Nauka: Nauka Moscow, [Russian] · Zbl 0057.07102
[48] Nenchu, G., Functional Anal. Appl., 19 (1985), Engl. transl. in
[49] Pavlov, B. S., Theory of extensions and solvable models, Uspekhi Mat. Nauk, 42, No. 6, 99-131 (1987), [Russian] · Zbl 0648.47010
[50] Saakyan, Sh. N., To the theory of resolvent of symmetric operator with infinite defect numbers, Dokl. Akad. Nauk Armyan. SSSR, 41, No. 4, 193-198 (1965), [Russian] · Zbl 0163.37804
[51] Szökefalvi-Nady, B.; Foias, C., Harmonic analysis of the operators in the space (1967), Academia Kiado: Academia Kiado Szeged-Bucharest
[52] Shvetsov, K. I., On Hamburger moment problem with additional supposision of absence of mass on given interval, Soobshch. Harkov. Mat. Obtshestva, 16 (1939), [Russian]
[53] Shmul’yan, Yu. L., Direct and inverse problems for resolvent matrices, Dokl. Acad. Nauk Ukrain. SSR Ser. A, No. 5, 514-517 (1970), [Russian] · Zbl 0211.44204
[54] Strauss, A. V., Generalized resolvent of symmetric operators, Izv. Akad. Nauk SSSR, Ser. Mat., 18, 51-86 (1954), [Russian]
[55] Strauss, A. V., The characteristic functions of the linear operators, Izv. Akad. Nauk SSSR Ser. Mat., 24, No. 1, 43-74 (1960), [Russian]
[56] Strauss, A. V., Math. USSR Izv., 2 (1968), Engl. transl. in
[57] Strauss, A. V., On extensions of semibounded operator, Dokl. Akad. Nauk SSSR, 211, No. 3, 543-546 (1973), [Russian]
[58] Fil’shtinskii, V. A., Moment problem on all axis, Khar’kov. Gos. Univ. Uchen. Zap. Mekh.-Mat. Fak. i Khar’kov. Mat. Obsh’ch., 30, 186-200 (1964), [Russian]
[59] Alonso, A.; Simon, B., The Birman-Krein-Vishok Theory of self-adjoint extensions of semibounded operators, Oper. Theory, 4, 251-270 (1980) · Zbl 0467.47017
[60] Ando, T.; Nischio, K., Positive selfadjoint extensions of positive symmetric operators, Tohoku Math. J., 2, No. 22, 65-75 (1970) · Zbl 0192.47703
[61] Calkin, I. W., Abstract symmetric boundary conditions, Trans. Amer. Math. Soc., 45, 369-442 (1939) · Zbl 0021.13802
[62] Kato, T., Perturbation Theory for Linear Operators (1966), Springer-Verlag: Springer-Verlag Berlin · Zbl 0148.12601
[63] Krein, M. G.; Langer, H., Uber die \(l\)-Function eines π-hermitesien Operators im Raume ΠK, Acta Sci. Math. Szeged, 34, 191-230 (1973) · Zbl 0276.47036
[64] Reed, M.; Synon, B., Methods of Modern Mathematical Physics II. Fourier Analysis, Self-adjointness (1975), Academic Press: Academic Press New York/London
[65] Atkinson, F. V., Discrete and Continuous Boundary Problems (1964), Academic Press: Academic Press New York/London · Zbl 0117.05806
[66] Vladimirov, V. S.; Zav’yalov, B. J., Automodel asymptotic of causal functions, Teor. Mat. Fiz., 50, No. 2, 163-194 (1982)
[67] Rofe-Beketov, F. S., On selfadjoint extensions of differential operators in a space of vector-valued functions, Teor. Funktsii Funktsional Anal. i Prilozhen, 8, 3-24 (1969), [Russian] · Zbl 0181.15401
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.