The powers of an operator of numerical radius one.

“…. , λ n ∈ D that are pseudohyperbolically well separated (see (7) below) and let δ denote a constant depending on the separation of the eigenvalues given in (8). Then in Theorem 2.2, we combine results from interpolation theory with von Neumann's inequality to deduce the existence of a constant M(δ) such that…”

Section: Resultsmentioning

confidence: 99%

“…The matrix C that we consider here appears in other work, including that of Crabb [7], Choi [5], and Greenbaum and Overton [24].…”

Section: Remark 310mentioning

confidence: 99%

Crouzeix’s Conjecture and Related Problems

Bickel

Gorkin

Greenbaum

et al. 2020

Comput. Methods Funct. Theory

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Crouzeix's conjecture asserts that, for any polynomial f and any square matrix A, the operator norm of f (A) satisfies the estimate f (A) ≤ 2 sup{| f (z)| : z ∈ W (A)}, (1) where W (A) := { Ax, x : x = 1} denotes the numerical range of A. This would then also hold for all functions f which are analytic in a neighborhood of W (A). We provide a survey of recent investigations related to this conjecture and derive bounds for f (A) for specific classes of operators A. This allows us to state explicit conditions that guarantee that Crouzeix's estimate (1) holds. We describe properties of related extremal functions (Blaschke products) and associated extremal vectors. The case where A is a matrix representation of a compressed shift operator is studied in some detail.

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“…Given an integer k with 2 ≤ k ≤ min(n, m + 1) , define the polynomial p ∈ P m by p(𝜁) = 𝜁 k−1 , set the matrix k ∈ M k to and set Ã = diag(𝛯 k , 0) ∈ M n . The matrix k was called the Choi-Crouzeix matrix of order k in [11], but after the paper was published, A. Salemi 1 informed us that it was introduced much earlier in a different context by Crabb [6]. The field of values of the Crabb matrix k is the unit disk, so the numerator of the Crouzeix ratio is 1, and p( Ã) = Ãk−1 = diag(𝛯 k−1 k , 0) is a matrix with just one nonzero, namely a 2 in the (1, k) position, so the denominator is 2; hence, the ratio is 0.5.…”

Section: The Crabb Matrix Conjectured Global Minimizers and Ice-cream-cone Stationary Pointsmentioning

confidence: 99%

Local minimizers of the Crouzeix ratio: a nonsmooth optimization case study

Overton

2021

Calcolo

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Given a square matrix A and a polynomial p, the Crouzeix ratio is the norm of the polynomial on the field of values of A divided by the 2-norm of the matrix p(A). Crouzeix's conjecture states that the globally minimal value of the Crouzeix ratio is 0.5, regardless of the matrix order and polynomial degree, and it is known that 1 is a frequently occurring locally minimal value. Making use of a heavy-tailed distribution to initialize our optimization computations, we demonstrate for the first time that the Crouzeix ratio has many other locally minimal values between 0.5 and 1. Besides showing that the same function values are repeatedly obtained for many different starting points, we also verify that an approximate nonsmooth stationarity condition holds at computed candidate local minimizers. We also find that the same locally minimal values are often obtained both when optimizing over real matrices and polynomials, and over complex matrices and polynomials. We argue that minimization of the Crouzeix ratio makes a very interesting nonsmooth optimization case study, illustrating among other things how effective the BFGS method is for nonsmooth, nonconvex optimization. Our method for verifying approximate nonsmooth stationarity is based on what may be a novel approach to finding approximate subgradients of max functions on an interval. Our extensive computations strongly support Crouzeix's conjecture: in all cases, we find that the smallest locally minimal value is 0.5.

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The powers of an operator of numerical radius one.

Cited by 13 publications

References 0 publications

Constants related to operators of class C?

Constants related to operators of class C?

Crouzeix’s Conjecture and Related Problems

Local minimizers of the Crouzeix ratio: a nonsmooth optimization case study

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