On unconditional bases of reproducing kernels in Fock-type spaces

The existence of unconditional bases of reproducing kernels in the Fock-type spaces $\mathcal F_{\varphi }$ with radial weights $\varphi $ is studied. It is shown that there exist functions $\varphi (r)$ of arbitrarily slow growth for which $\ln r=o(\varphi (r))$ as $r\to\infty$ and there are no unconditional bases of reproducing kernels in the space $\mathcal F_{\varphi }$. Thus, a criterion for the existence of unconditional bases cannot be given only in terms of the growth of the weight function.