## A note on Riesz bases of eigenvectors of certain holomorphic operator-functions.(English)Zbl 0972.47011

The operator-function ${\mathcal A}(\lambda)=A-\lambda I+Q(\lambda)$ is considered acting in a separable Hilbert space, where $$A$$ is an operator with compact resolvent, $$Q(\lambda)$$ is an A-compact linear operator for each $$\lambda\in\Omega$$ ($$\Omega$$ is an open connected subset of $$C$$). It is assumed that all the eigenvalues $$\{\lambda_j\}_1^{\infty}$$ of $$A$$ beyond a certain index are all simple, lie in $$\Omega$$, and the corresponding eigenvectors form a Riesz basis of finite defect. The main theorem says that if there are bounds for $$||Q(\lambda)(A-\lambda I)^{-1}||$$ on certain circles around the $$\lambda_j$$ which decrease rapidly enough with $$j$$, then the spectrum of $${\mathcal A}$$ in each circle consists only of a single eigenvalue and the corresponding eigenvectors also form a Riesz basis of finite defect.

### MSC:

 47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces 46A35 Summability and bases in topological vector spaces 47B06 Riesz operators; eigenvalue distributions; approximation numbers, $$s$$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
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### References:

 [1] Adamjan, V. M.; Langer, H., Spectral properties of a class of rational operator valued functions, J. Operator Theory, 33, 259-277 (1995) · Zbl 0841.47010 [2] Bogomolova, J. B., Some questions of the spectral analysis of a nonselfadjoint differential operator with a “floating singularity” in the coefficient, Differentsial’nye Uravneniya, 21, 1843-1849 (1985) · Zbl 0658.34017 [3] Friedman, A.; Shinbrot, M., Nonlinear eigenvalue problems, Acta Math., 121, 77-125 (1968) · Zbl 0162.45704 [4] Gohberg, I. C.; Goldberg, S.; Kaashoek, M. A., Classes of Linear Operators, Vol. I. Classes of Linear Operators, Vol. I, Operator Theory: Advances and Applications, 29 (1990), Birkhäuser: Birkhäuser Basel · Zbl 0745.47002 [5] Gohberg, I. C.; Kaashoek, M. A.; Lay, D. C., Equivalence, linearization, and decomposition of holomorphic operator functions, J. Funct. Anal., 28, 102-144 (1978) · Zbl 0384.47018 [6] Gohberg, I. C.; Krein, M. S., Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space. Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, Translations of Mathematical Monographs, 18 (1969), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0181.13504 [7] Gohberg, I. C.; Sigal, E. I., An operator generalization of the logarithmic residue theorem and the theorem of Rouché, Math. USSR-Sb., 13, 603-625 (1971) · Zbl 0254.47046 [8] Griesemer, M.; Lutgen, J., Accumulation of discrete eigenvalues of the radial Dirac operator, J. Funct. Anal., 162, 120-134 (1999) · Zbl 0926.34074 [9] Kato, T., Perturbation Theory for Linear Operators. Perturbation Theory for Linear Operators, Grundlehren Math. Wiss., 132 (1976), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0342.47009 [10] Langer, H.; Mennicken, R.; Möller, M., A second order differential operator depending nonlinearly on the eigenvalue parameter, Operator Theory: Advances and Applications (1990), Birkhäuser: Birkhäuser Basel, p. 319-332 · Zbl 0734.34070 [11] Langer, H.; Tretter, Ch., Spectral decomposition of some nonselfadjoint block operator matrices, J. Operator Theory, 39, 339-359 (1998) · Zbl 0996.47006 [12] Levitan, B. M.; Sargsjan, I. S., Sturm-Liouville and Dirac Operators (1991), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0302.47036 [13] Linden, H., Linearization, completeness, and spectral asymptotics for certain rational and meromorphic operator functions, J. Operator Theory, 39, 219-247 (1998) · Zbl 0998.47006 [14] J. Lutgen, Boundary Value Problems for Some Ordinary Differential Equations Depending Nonlinearly on the Spectral Parameter, Dissertation, Universität Regensburg, 1997.; J. Lutgen, Boundary Value Problems for Some Ordinary Differential Equations Depending Nonlinearly on the Spectral Parameter, Dissertation, Universität Regensburg, 1997. [15] Lutgen, J., Eigenvalue accumulation for singular Sturm-Liouville problems nonlinear in the spectral parameter, J. Differential Equations, 159, 515-542 (1999) · Zbl 0951.34058 [16] Magnus, R., The spectrum and eigenspaces of a meromorphic operator-valued function, Proc. Roy. Soc. Edinburgh Sect. A, 127, 1027-1051 (1997) · Zbl 0885.30028 [17] Markus, A. S., A basis of root vectors of a dissipative operator, Soviet Math. Dokl., 1, 599-602 (1960) · Zbl 0095.31203 [18] Markus, A. S., Expansions in root vectors of a slightly perturbed self-adjoint operator, Soviet Math. Dokl., 3, 104-108 (1962) · Zbl 0122.11602 [19] Markus, A. S., Introduction to the Spectral Theory of Polynomial Operator Pencils. Introduction to the Spectral Theory of Polynomial Operator Pencils, Translations of Mathematical Monographs, 71 (1986), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0678.47006 [20] Markus, A. S.; Matsaev, V. I., Comparison theorems for spectra of linear operators and spectral asymptotics, Trans. Moscow Math. Soc., 45, 139-187 (1984) · Zbl 0557.47009 [21] Müller, P. Heinz, Eine neue Methode zur Behandlung nichtlinearer Eigenwertaufgaben, Math. Z., 70, 381-406 (1959) · Zbl 0083.34901 [22] Shinbrot, M., Note on a nonlinear eigenvalue problem, Proc. Amer. Math. Soc., 14, 552-558 (1963) · Zbl 0115.33802 [23] Shinbrot, M., A nonlinear eigenvalue problem, II, Arch. Rational Mech. Anal., 15, 368-376 (1964) · Zbl 0127.33702 [24] Triebel, H., Höhere Analysis (1980), Verlag Harri Deutsch · Zbl 0455.47035 [25] Turner, R. E., Some variational principles for a nonlinear eigenvalue problem, J. Math. Anal. Appl., 17, 151-160 (1967) · Zbl 0147.12201 [26] Weidmann, J., Linear Operators in Hilbert Spaces. Linear Operators in Hilbert Spaces, Graduate Texts in Mathematics, 68 (1980), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0434.47001 [27] Weinberger, H. F., On a nonlinear eigenvalue problem, J. Math. Anal. Appl., 21, 506-509 (1968) · Zbl 0162.45801
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