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An estimate for the number of eigenvalues of a Hilbert-Schmidt operator in a half-plane. (English) Zbl 07225495
Summary: Let \(A\) and \(\tilde{A}\) be Hilbert-Schmidt operators. For a constant \(r>0\), let \(i_+(r, A)\) be the number of the eigenvalues of \(A\) taken with their multiplicities lying in the half-plane \(\{z\in\mathbb{C}: \mathfrak{R} z>r\}\). We suggest the conditions that provide the equality \(i_+(r, \tilde{A})=i_+(r, A)\).
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
15A18 Eigenvalues, singular values, and eigenvectors
15A69 Multilinear algebra, tensor calculus
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[1] B. Bilir and C. Chicone, A generalization of the inertia theorem for quadratic matrix polynomials, Linear Algebra and its Applications 280 (1998) 229-240. · Zbl 0934.15020
[2] B.E. Cain, Inertia theory, Linear Algebra Appl. 30 (1980) 211-240. · Zbl 0441.47030
[3] D. Carlson and H. Schneider, Inertia theorems: the semidefinite case, J. Math. Anal. Appl. 6 (1963) 430-446. · Zbl 0192.13402
[4] Chi-Tsong Chen, Inertia theorem for general matrix equations, J. Math. Anal. Appl. 49 (1975), 207-210. · Zbl 0351.15010
[5] B.N. Datta, Stability and inertia, Linear Algebra Appl. 302/303 (1999), 563-600. · Zbl 0972.15009
[6] M.I. Gil’, Bounds for the spectrum of a two parameter matrix eigenvalue problem, Linear Algebra and its Appl. 498 (2016), 201-218. · Zbl 1334.15029
[7] M.I. Gil’, Resolvents of operators on tensor products of Euclidean spaces, Linear and Multilinear Algebra 64 (2016), no. 4, 699-716. · Zbl 1345.15005
[8] M.I. Gil’, Operator Functions and Operator Equations, World Scientific, New Jersey, 2018. · Zbl 1422.47004
[9] M.I.Gil’, Conservation of the number of eigenvalues of finite dimensional and compact operators inside and outside circle, Functional Analysis, Approximation and Computation 10(2) (2018), 47-54. · Zbl 1392.15017
[10] R.A. Horn and Ch.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991. · Zbl 0729.15001
[11] L. Lerer, I. Margulis and A. Ran, Inertia theorems based on operator Lyapunov equations, Operators and Matrices 2 (2008), no. 2, 153-166. · Zbl 1163.47010
[12] R. Loewy, An inertia theorem for Lyapunov’s equation and the dimension of a controllability space, Linear Algebra Appl. 260 (1997), 1-7. · Zbl 0894.93018
[13] A. Ostrowski and H. Schneider, Some theorems on the inertia of general matrices, J. Math. Anal. Appl. 4(l) (1962), 72-84. · Zbl 0112.01401
[14] A.J. Sasane and R.F. Curtain, Inertia theorems for operator Lyapunov inequalities, Systems and Control Letters 43 (2001), 127-132. · Zbl 0974.93026
[15] T. Stykel, Stability and inertia theorems for generalized Lyapunov equations, Linear Algebra and its Applications 355 (2002), 297-314. · Zbl 1016.15010
[16] H.K. Wimmer, On the Ostrowski-Schneider inertia theorem, J. Math. Anal. Appl. 41 (1973), 164-169. · Zbl 0251.15011
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