TY - JOUR

T1 - Crouzeix’s Conjecture and Related Problems

AU - Bickel, Kelly

AU - Gorkin, Pamela

AU - Greenbaum, Anne

AU - Ransford, Thomas

AU - Schwenninger, Felix L.

AU - Wegert, Elias

PY - 2020/11

Y1 - 2020/11

N2 - 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)‖≤2sup{|f(z)|:z∈W(A)},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.

AB - 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)‖≤2sup{|f(z)|:z∈W(A)},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.

KW - Blaschke products

KW - Crouzeix’s conjecture

KW - Nevanlinna–Pick interpolation

KW - Numerical radius

KW - Numerical range

KW - Shift operator

KW - 22/2 OA procedure

UR - http://www.scopus.com/inward/record.url?scp=85092703437&partnerID=8YFLogxK

U2 - 10.1007/s40315-020-00350-9

DO - 10.1007/s40315-020-00350-9

M3 - Article

AN - SCOPUS:85092703437

VL - 20

SP - 701

EP - 728

JO - Computational Methods and Function Theory

JF - Computational Methods and Function Theory

SN - 1617-9447

IS - 3-4

ER -