The \(q\)-numerical radius of weighted shift operators with periodic weights
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 \(w_q(A)= s_{i_1}\) is also given when \(s_{i_1}=\cdots= s_{i_p}> s_j\) for \(1\leq i_1<\cdots< i_p\leq m\), \(p\geq 2\) and all \(j\neq i_1,\dots, i_p\). It is shown, for a unilateral weighted shift, denoted by \([s_1,s_2,s_3,s_4]\), with 4-periodic weights \(\{s_1,s_2,s_3, s_4\}\) satisfying \(s_4\geq\cdots\geq s_1\geq 0\), that \[ w_q([s_2, s_4,s_3,s_1])\geq w_q([s_1, s_4, s_3, s_2])\geq w_q([s_1, s_4, s_2, s_3]). \] The paper concludes with a lower bound of \(w_q(A)\) for \(A\) a unilateral weighted shift with periodic weights.
scientific article; zbMATH DE number 5132624
scientific article