Constants related to powers of ρ-contractions

Abstract

Let A be a bounded linear operator on a Hilbert space H. In this paper, we show that if A is a numerical contraction and $1\le n<\infty$, then $\Vert Ax\Vert =\Vert A^{2}x\Vert =\cdots =\Vert A^{n}x\Vert =\sqrt{2(n+1)/n}$ for some unit vector $x\in H$ if and only if A is unitarily similar to an operator of the form $A_n\oplus D$, where D is a numerical contraction and$A_n=\left[\begin{array}{ccccc}0& & & & \\ \sqrt{\frac{2(n+1)}{n}}& 0& & & \\ & 1& 0& & \\ & & \ddots & \ddots & \\ & & & 1& 0\end{array}\right]\in M_{n+1}.$ Moreover, we also show that if $\rho >1$ and A is a ρ-contraction, then ${\lim }_n\Vert A^{n}x\Vert =\sqrt{\rho }$ for some unit vector $x\in H$ if and only if A is unitarily similar to an operator of the form $A_{\rho ,\infty }\oplus D$, where D is a ρ-contraction and$A_{\rho ,\infty }=\left[\begin{array}{ccccc}0& & & & \\ \sqrt{\rho }& 0& & & \\ & 1& 0& & \\ & & 1& 0& \\ & & & \ddots & \ddots \end{array}\right]\text{ on }{\ell }^2.$