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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 $$S(\phi)$$.

##### MSC:
 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
##### Keywords:
numerical range; numerical radius; compression of the shift
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