Convexity characteristic of Calderón-Lozanovskiĭ sequence spaces (Q2922210)

Recall that the characteristic of convexity of a Banach space \(X=(X,\|\cdot\|)\) is defined by the formula NEWLINE\[NEWLINE\varepsilon_{0}(X)=\sup\{\varepsilon\in [0,2):\delta_X(\varepsilon)=0\},NEWLINE\]NEWLINE where \(\delta_X(.)\) is the modulus of convexity of \(X\), that is, \(\delta_X(.)\) is the function from \([0,2]\) into \([0,1]\) given by the formula NEWLINE\[NEWLINE\delta_X(\varepsilon)=\inf\left\{1-\left\|\frac{x+y}{2}\right\|:\|x\|=\|y\|=1\wedge\|x-y\|\geq\varepsilon\right\}.NEWLINE\]NEWLINE Given a Köthe sequence space \(e\) and an Orlicz function \(\Phi\), let us denote by \(e_\Phi\) the Calderón-Lozanovskiĭ space that consists of all real sequences \(x=(x_n)_{n=1}^\infty\) such that the sequence \(\Phi\circ \lambda x=(\Phi(\lambda x_n))_{n=1}^\infty\) belongs to \(e\) for some \(\lambda>0\) and let the norm of \(x\in e_\Phi\) be defined as \(\|x\|_{e_\Phi}=\inf\{\lambda>0:\rho(\frac{x}{\lambda})\leq 1\}\), where NEWLINE\[NEWLINE\rho(x)=\begin{cases} \|\Phi\circ x\|_e &\text{if }\Phi\circ x\in e,\\ \infty & \text{otherwise.}\end{cases}NEWLINE\]NEWLINE Under the assumption that the Orlicz function \(\Phi\) is strictly convex on the interval \([0,\mu_b]\), where \(\mu_b=\Phi^{-1}\left(\frac{1}{\|\chi_{\{1\}}\|_e}\right)\), the characteristic of convexity of the space \(e_\Phi\) is calculated. This general result is applied to calculate the characteristic of Orlicz-Lorentz sequence spaces \(\lambda_{\Phi,\omega}\) in some special cases. In cases when this characteristic is not calculated, an upper estimate is given.