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Numerical Range of Matrix Polynomials
SIAM Journal on Matrix Analysis and Applications, 1994Let \(A_ i\), \(i = 1, \dots, m\) be \(n \times n\) matrices with complex coefficients and consider the matrix polynomial \(P(\lambda) = \sum_{i=0}^ m A_ i \lambda^ i\). The numerical range of \(P(\lambda)\) is defined through \[ W \bigl( P(\lambda) \bigr) : = \{\mu \in \mathbb{C} \mid x^* P(\mu)x = 0 \quad \text{for some nonzero} \quad x \in \mathbb{C}
Chi-Kwong Li, Leiba Rodman
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Compressions and Dilations of Numerical Ranges
SIAM Journal on Matrix Analysis and Applications, 1999The authors continue their study of the numerical range \(\text{NR} [A]\) of an \(n\times n\) complex matrix \(A\). They express \(\text{NR} [A]\) as the union of the numerical ranges of \(k\times k\) matrices for \(2\leq k< n\). In this way each set in the union can be considered as a dilation of \(\text{NR} [A]\). They also consider compressions of \(
John Maroulas, Maria Adam
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Ratio Numerical Ranges of Operators
Integral Equations and Operator Theory, 2011Let \(H\) be a Hilbert space and \(L(H)\) the algebra of bounded linear operators on \(H\). For \(A\in L(H)\), the numerical range of \(A\) is given by \(W(A)= \{\langle Ax,x\rangle:x\in H,\;\langle x,x\rangle=1\}\). For \(A,B\in L(H)\), \(B\neq 0\), define the ratio numerical range \[ W(A/B)=\left\{\frac{\langle Ax,x\rangle}{\langle Bx,x\rangle}:x\in ...
Rodman, Leiba, Spitkovsky, Ilya M.
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Normality and the Higher Numerical Range
Canadian Journal of Mathematics, 1978Let Mn(C) be the vector space of all w-square complex matrices. Denote by (• , •) the standard inner product in the space C n of complex n-tuples. For a matrix A ∈ Mn(C) and an n-tuple c = (c1,…
Marcus, Marvin +2 more
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On a Family of Generalized Numerical Ranges
Canadian Journal of Mathematics, 1974Throughout this note, an operator will always mean a bounded linear operator acting on a Hilbert space X into itself, unless otherwise stated. The class Cρ (0 < ρ < ∞ ) of operators, considered by Sz.-Nagy and Foiaş [5], is defined as follows: An operator T is in Cρ if Tnx = pPUnx for all x ∊ X, n = 1, 2, . . .
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The numerical range of a Toeplitz operator
Proceedings of the American Mathematical Society, 1972In this paper we explicitly compute the numerical range of an arbitrary Toeplitz operator on the classical Hardy space H 2
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The C -Numerical Range in Infinite Dimensions
Linear and Multilinear Algebra, 2020G Dirr, Frederik vom Ende
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Product of operators and numerical range
Linear and Multilinear Algebra, 2016Mao-Ting Chien +2 more
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