The \(q\)-numerical radius of weighted shift operators with periodic weights
type: Item
instance of: scholarly article
title: The \(q\)-numerical radius of weighted shift operators with periodic weights
zbMATH Open document ID: 1117.15027
author: Mao-Ting Chien
author: Hiroshi Nakazato
published in: Linear Algebra and its Applications
publication date: 2007-03-09T00:00:00Z
review text: For a bounded linear operator \(T\) on a complex Hilbert space with inner product \(\langle.,.\rangle\) and for \(0\leq q\leq 1\), the \(q\)-numerical range of \(T\) is, by definition, \(W_q(T)= \{\langle Tx, y\rangle:\| x\|= \| y\|= 1\), \(\langle x,y\rangle= q\}\), and the \(q\)-numerical radius of \(T\) is \(q_q(T)= \sup\{|z|: z\in W_q(T)\}\). If \(q= 1\), then \(W_q(T)\) and \(w_q(T)\) are exactly the classical numerical range \(W(T)\) and numerical radius \(w(T)\), respectively. Let \(A\) be a unilateral (resp. bilateral) weighted shift on \(\ell^2(\mathbb{N})\) (resp., \(\ell^2(\mathbb{Z})\)) with nonnegative weights \(\{s_n\}^\infty_{n=1}\) (resp., \(\{s_n\}^\infty_{n=-\infty}\)): \(A(x_1,x_2,\dots)= (0, s_1 x_1, s_2x_2,\dots)\) (resp., \(A(\dots, x_{-1},\underline{x_0}, x_1,\dots)= (\dots, s_{-2} x_{-2}, \underline{s_{-1} x_{-1}}, s_0 x_0,\dots)\)). It is known that \(W_q(T)\) is a circular disc centered at the origin with radius given by Tsing's circular union formula. The latter involves the height function \(h(z)= \sup\{t: (z,t)\in W(A, A^*A)\}\), where \(W(A, A^* A)= \{(\langle Ax, x\rangle, \langle A^* Ax,x\rangle):\| x\|= 1\}\) is the Davis-Wielandt shell of \(A\). The main purpose of this paper is to study \(w_q(A)\) with \(A\) periodic weights \(\{s_1,\dots, s_m,s_1,\dots, s_m,\dots\}\) (resp., \(\{\dots, s_1,\dots, s_m,s_1,\dots, s_m,\dots\}\)). It is shown, by relating the height functions of \(A\) and \(S\), that \(w_q(A)= w_q(S)\), where \(S\) is the \(m\)-by-\(m\) matrix \[ \left[\begin{matrix} 0 &&& s_m\\ s_1 & \ddots\\ &\ddots\\ && s_{m-1} & 0\end{matrix}\right]. \] Although a general formula for \(w_q(A)\) is yet unknown, the authors obtain, by using this, one for \(A\) with 2-periodic weights: \[ w_q(A)= {s_1+ s_2\over 2}+ \sqrt{1- q^2} {|s_1- s_2|\over 2}. \] For \(m\geq 3\), if \(s_j< s_2\) for all \(j\neq 2\), then a formula for \(w_q(A)\) is also obtained for a sufficiently small. In this case, a characterization for \(w_q(A)= s_{i_1}\) …
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 47A12
Mathematics Subject Classification ID: 47B37
zbMATH DE Number: 5132624
zbMATH Keywords: \(q\)-numerical range
zbMATH Keywords: unilateral weighted shift
zbMATH Keywords: bilateral weighted shift
zbMATH Keywords: operators on sequence spaces
MaRDI profile type: Publication
full work available at URL: j.laa.2006.09.017
OpenAlex ID: W2060952175
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DOI: 10.1016/J.LAA.2006.09.017
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description: scientific article; zbMATH DE number 5132624