Numerical Range and Poncelet Property

Abstract

In this survey article, we give an expository account of the recent developments on the Poncelet property for numerical ranges of the compressions of the shift S(Á). It can be considered as an updated and more advanced edition of the recent expository article published in the American Mathematical Monthly by the second author on this topic. The new information includes:(1) a simplified approach to the main results (generalizations of Poncelet, Brianchon-Ceva and Lucas-Siebeck theorems) in this area, (2) the recent discovery of Mirman refuting a previous conjecture on the coincidence of Poncelet curves and boundaries of the numerical ranges of finite-dimensional S(Á), and (3) some partial generalizations by the present authors of the above-mentioned results from the unitary-dilation context to the normal-dilation one and also from the finite-dimensional S(Á) to the infinite-dimensional.

Ellipses and compositions of finite Blaschke products

“…Thus a Blaschke product B is decomposable with an inside factor of degree 2 if for some zero a 2 of B, the transformation T a 2 (z) = a 2 −z 1−a 2 z permutes the zeros of B. Equivalently, B is decomposable if there exists an ordering of the zeros of B such that a 2k−1 = a 2 −a 2k 1−a 2 a 2k for all 1 ≤ k ≤ n. We now apply results of Gau and Wu (see [7,13]) as well as techniques above to give a new proof of Fujimura's theorem, provide information on the location of the zeros, and to clarify the connection to the numerical range of the compression of the shift. Proof.…”

“…yields the following matrix representation for S B [7] in the case of a degree-3 Blaschke product B with zeros a, b and c in D, which we henceforth denote by A B ; that is:…”

Poncelet's theorem, Sendov's conjecture, and Blaschke products

“…We now compare Theorem 2 to the following. The connection between these two theorems has been well studied; in particular, Gau and Wu's work [7][8][9][10]26] and Mirman's papers [18][19][20][21][22] tie these theorems together using Poncelet's theorem.…”

“…In fact, in this case, A is unitarily equivalent to the n × n Jordan block (see [9,Theorem 5.11] and [7, Theorem 4.5]). It was pointed out to us that Corollary 16 follows from this fact with n = 2 and [7, Lemma 4.2].…”

Poncelet's theorem, paraorthogonal polynomials and the numerical range of compressed multiplication operators

“…The final theme concerns a class of finite dimensional matrices, now called S n , studied in a series of independent papers by Gau-Wu [24,25,26,27,28,29,30], also [76], and by Mirman [50,51,52,53,54] both series starting in 1998. Recall that an operator on a Hilbert space is called a contraction if its norm is at most 1.…”