A sharpened Schwarz-Pick operatorial inequality for nilpotent operators

Abstract

Let denote by S(φ) the extremal operator defined by the compression of the unilateral shift S to the model subspace H(φ) = H 2 ⊖ φ H 2 as the following S(φ)f (z) = P (zf (z)), where P denotes the orthogonal projection from H 2 onto H(φ) and φ is an inner function on the unit disc. In this mathematical notes, we give an explicit formula of the numerical radius of S(φ) in the particular case where φ is a finite Blaschke product with unique zero and an estimate on the general case. We establish also a sharpened Schwarz-Pick operatorial inequality generalizing a U. Haagerup and P. de la Harpe result for nilpotent operators [16].

Blaschke Products, Level Sets, and Crouzeix's Conjecture

“…A lot is known about the numerical ranges W (S Θ ) associated to unicritical Θ. For example in [22], Gaaya characterized their numerical radii and established a number of intermediate results, including the following useful equality in his Proposition 2.6:…”

“…For now, recall that Crouzeix's conjecture states: given a square matrix A, the best constant C for which (22) p(A) ≤ C max Before proceeding to the proof, a few comments are in order. First in [14], Crouzeix proved that the numerical range of a 3×3 nilpotent matrix is a 2-spectral set; that is, in this case the constant C in (22) can be taken to be 2. Because A t from ( 12) is nilpotent, that establishes Proposition 5.6 but with constant 2.…”

“…In those situations, there is a range of t-values (i.e. a range for the modulus of the zero of the unicritical Θ) where the constant in (22) with A = S Θ is less than 2. For the proofs, we require the following remark.…”

“…The statement of ( 2) suggests that we need to study the numerical range of the associated compression of the shift S Θ . For unicritical Θ, these numerical ranges were studied by Gaaya in [21,22], Gau and Wu in [27,28], and in work of Partington and the second author [30]. When deg Θ = 3, results about W (S Θ ) are encoded in Crouzeix's work [14]; indeed, his arguments imply that the full Crouzeix conjecture holds for S Θ when deg Θ = 3 and Θ is unicritical.…”

Zero-dilation Indices of KMS Matrices

“…The study of the numerical range of J n (a) was started by Gaaya in [1,2] and continued by the present authors in [5]. As a meeting ground of the classes of nilpotent, Toeplitz, nonnegative, S n -and S −1 n -matrices, J n (a) has diverse and interesting properties concerning its numerical range.…”