Đoković, Dragomir Ž. Quadratic cones invariant under some linear operators. (English) Zbl 0619.15022 SIAM J. Algebraic Discrete Methods 8, 186-191 (1987). A quadratic cone in a finite dimensional space is the set of x satisfying f(x,x)\(\geq 0\), where f in an indefinite Hermitian form. The author characterizes the linear operators which have such a fixed cone invariant. He shows furthermore that if the spectral radius and the norm of an operator A coincide for a submultiplicative norm on the operators then there exists an A-invariant quadratic cone of specified signature. Reviewer: T.B.Andersen Cited in 2 Documents MSC: 15A63 Quadratic and bilinear forms, inner products 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 15A18 Eigenvalues, singular values, and eigenvectors Keywords:indefinite Hermitian form; spectral radius; submultiplicative norm; invariant quadratic cone PDFBibTeX XMLCite \textit{D. Ž. Đoković}, SIAM J. Algebraic Discrete Methods 8, 186--191 (1987; Zbl 0619.15022) Full Text: DOI References: [1] Dieudonné, J., Foundations of modern analysis, (1969) · Zbl 0176.00502 [2] Djoković, DragomirŽ., Extreme rays of certain cones of Hermitian forms, Proc. Amer. Math. Soc., 83, 243, (1981) · Zbl 0474.15009 [3] Friedland, Shmuel, A characterization of transform absolute norms, Linear Algebra Appl., 28, 63, (1979) · Zbl 0421.15023 · doi:10.1016/0024-3795(79)90119-8 [4] Givens, Wallace, Fields of values of a matrix, Proc. Amer. Math. Soc., 3, 206, (1952) · Zbl 0048.25003 [5] Householder, AlstonS., The theory of matrices in numerical analysis, (1964) · Zbl 0161.12101 [6] Krein, M. G.; Šmul’jan, Ju. L., On the plus-operators in a space with an indefinite metric, Mat. Issled., 1, 131, (1966) · Zbl 0201.16802 [7] Loewy, Raphael; Schneider, Hans, Positive operators on the {\it n}-dimensional ice cream cone, J. Math. Anal. Appl., 49, 375, (1975) · Zbl 0308.15011 · doi:10.1016/0022-247X(75)90186-9 [8] Mott, J. L.; Schneider, Hans, Matrix algebras and groups relatively bounded in norm, Arch. Math., 10, 1, (1959) · Zbl 0201.36901 [9] Norms and numerical ranges infinite dimensionsunpublished [10] Schneider, Hans; Vidyasagar, Mathukumalli, Cross-positive matrices, SIAM J. Numer. Anal., 7, 508, (1970) · Zbl 0245.15008 · doi:10.1137/0707041 [11] Vandergraft, JamesS., Spectral properties of matrices which have invariant cones, SIAM J. Appl. Math., 16, 1208, (1968) · Zbl 0186.05701 · doi:10.1137/0116101 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.