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On the numerical radius of the truncated adjoint shift. (English) Zbl 1244.47003
For any nonconstant inner function \(\phi\), let \(S(\phi)\) denote the compression of the unilateral shift on the Hardy space \(H^2\) of the unit disc to the subspace \(H(\phi)= H^2\ominus\phi H^2\): \[ (S(\phi)f)(z)= P_{H(\phi)}(zf(z)), \] where \(P_{H(\phi)}\) denotes the (orthogonal) projection from \(H^2\) onto \(H(\phi)\). Such operators were first studied by D. Sarason in 1967. It is known that \(\dim H(\phi)= n<\infty\) exactly when \(\phi\) is a Blaschke product with \(n\) zeros: \[ \phi(z)= \prod^n_{j=1} (z- a_j)/(1-\overline a_j z), \] where \(a_j\) is in \(\mathbb{D}\equiv \{z\in\mathbb{C}: |z|< 1\}\) for all \(j\).
In Section 2 of this paper, the author proves (in Theorem 2.1) that if \(F(z)= p(z)/q(z)\), where \(p\) and \(q\) are polynomials with no common zero, is such that \(F(e^{it})\geq 0\) for all real \(t\), and \[ \phi(z)= z^{\max\{0,m- n+1\}} \prod^q_{j=1} {z- b_j\over 1-\overline b_jz}, \] where the \(b_j\)’s are the distinct nonzero roots of \(q= 0\) in \(\mathbb{D}\), and \(m\) (resp., \(n\)) is the number of nonzero roots of \(p=0\) (resp., \(q=0\)) in \(\mathbb{D}\) counting multiplicity, then \(|c_k|\leq c_0 w(S(\phi)^k)\) for all \(k\geq 1\), where \(c_k\) (resp., \(c_0\)) is the \(k\)th (resp., \(0\)th) coefficient of the Laurent series expansion of \(F(z)\), \(F(z)=\sum^\infty_{j=-\infty} c_j z^j\), and \(w(.)\) denotes the numerical radius of an operator. This is an improvement of a result of C. Badea and G. Cassier [Adv. Math. 166, No. 2, 260–297 (2002; Zbl 1049.47004)] for \(m< n\), which in turn generalizes the ones by U. Haagerup and P. de la Harpe [Proc. Am. Math. Soc. 115, No. 2, 371–379 (1992; Zbl 0781.47014)] and T. Okayasu and Y. Ueta [Sci. Math. Jpn. 56, No. 1, 115–122 (2002; Zbl 1030.42005)]. It can even be traced back to E. von Egerváry and O. Szász [Math. Z. 27, 641–652 (1928; JFM 54.0314.01)] and L. Fejér [J. Reine Angew. Math. 146, 53–82 (1915; JFM 45.0406.02)].
One of the main results in Section 3 is Proposition 3.2, which determines the numerical radius of the matrix \[ J_n(\alpha)= \left(\begin{matrix} 0 & \alpha &\cdots & \alpha^{n-1}\\ &0 &\ddots &\vdots\\ &&\ddots & \alpha\\ 0 &&&0\end{matrix}\right) \] with \(0\leq \alpha< 1\). This is then used (in Theorem 3.7) to derive a lower bound of the numerical radius of \(S(\phi)\) for \(\phi(z)= ((z- a)/(1-\overline az))^n\), \(|a|< 1\). This, together with an upper bound of \(w(S(\phi))\), is proven via a standard matrix representation of \(S(\phi)\).

47A12 Numerical range, numerical radius
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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