Between the von Neumann inequality and the Crouzeix conjecture (Q2197272)

An observation due to [\textit{B. Sz. -Nagy} et al., Harmonic analysis of operators on Hilbert space. New York, NY: Springer (2010; Zbl 1234.47001), Theorem 11.1] that a bounded operator \( T \) in a Hilbert space \( H \) has a \( \rho \)-dilation, \(\rho>0\), if and only if \[ \left( 2 \rho^{-1 } - 1 \right) \| z T h \|^2 + \left( 2 - 2\rho^{-1 } \right) \Re ( z T h , h ) \le \| h \|^2 \] for all \( h \in H \) and \( z \in \mathbb C \), \( |z| \le 1 \). In particlular, \( T \) has a 2-dilation if and only if its the numerical range, \( W ( T ) = \{ ( Th , h ), h \in H, \| h \| = 1 \} \), is contained in the closed unit disc \( \overline {\mathbb D} \). In the paper authors define and study a deformation of the numerical range, \( W^\rho ( T ) \), \( \rho \ge 1 \), suggested by a form of the estimate above, for which the existence of a \( \rho \)-dilation is equivalent to the inclusion \( W^\rho ( T ) \subset \overline {\mathbb D} \). The set \( W^2 ( T ) \) coincides with the closure of \( W ( T ) \). The spectral constant \( \Psi_\rho (T ) \) for the sets \( W^\rho ( T ) \) is shown to be continuous and monotone as a function of \( \rho \in [ 1 , 2 ] \). A corollary is a reformulation of the Crouzeix conjecture in terms of \( \Psi_\rho ( T ) \).