Garcia, Stephan Ramon Approximate antilinear eigenvalue problems and related inequalities. (English) Zbl 1130.47007 Proc. Am. Math. Soc. 136, No. 1, 171-179 (2008). A conjugate linear operator \(C\) in a separable complex Hilbert space \(H\) is called conjugation if \(C^2=I\) and \(\langle Cy,Cx \rangle=\langle x,y\rangle\). A bounded operator \(T\) in \(H\) is said to be \(C\)-symmetric if \(T=CT^*C\). It is called complex symmetric if it is \(C\)-symmetric for some conjugation \(C\) [cf.S.R.Garcia and M.Putinar, Trans.Am.Math.Soc.358, No.3, 1285–1315 (2006; Zbl 1087.30031)].A classical result in matrix theory asserts that the singular values of an \(n \times n\) complex symmetric matrix \(A\) can be characterised as the nonnegative solutions \(\lambda\) to the antilinear eigenvalue problem \(Ax=\lambda\bar{x}\), where \(x \neq 0\) and \(\bar{x}\) denotes entry-by-entry complex conjugation of a vector \(x \in {\mathbb C}^n\) [cf.R.A.Horn and C.R.Johnson, “Matrix analysis” (Cambridge University Press) (1985; Zbl 0576.15001)].In the paper under review, the author proves that if \(T\) is a \(C\)-symmetric operator and \(\lambda \geq 0\), then \(\lambda\) belongs to the spectrum \(\sigma(| T|)\) of \(| T|=(T^*T)^{1/2}\) if and only if there exists a sequence \(\{f_n\}\) of unit vectors such that \(\lim_{n\to\infty}\|(T-\lambda C)f_n\|=0\). Using a complex symmetric block technique, the author then derives a system of general inequalities restricting the possible location of spectral values \(\lambda\in\sigma(| T|)\) for an arbitrary bounded linear operator \(T\) in \(H\). He also gives some sharp inequalities for the operator norm. Reviewer: Mohammad Sal Moslehian (Mashhad) Cited in 5 Documents MSC: 47A30 Norms (inequalities, more than one norm, etc.) of linear operators 47A75 Eigenvalue problems for linear operators 47A10 Spectrum, resolvent Keywords:complex symmetric operator; operator norm; triangle inequality; selfadjoint operator; Cartesian decomposition; approximate antilinear eigenvalue problem Citations:Zbl 1087.30031; Zbl 0576.15001 PDFBibTeX XMLCite \textit{S. R. Garcia}, Proc. Am. Math. Soc. 136, No. 1, 171--179 (2008; Zbl 1130.47007) Full Text: DOI References: [1] Peter Arbenz and Michiel E. Hochstenbach, A Jacobi-Davidson method for solving complex symmetric eigenvalue problems, SIAM J. Sci. 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