Let \(f(z)=\sum_\nu f_\nu z^\nu\) be a holomorphic function on the unit ball \({\mathbb B}^n\) in \({\mathbb C}^n\). For \(\alpha\in{\mathbb R}\), \textit{R.-H. Zhao} and \textit{K. Zhu} [Mém. Soc. Math. Fr., Nouv. Sér. 115, 1--103 (2008; Zbl 1176.32001)] considered \(\|f\|_{\alpha,\#}^2:=\sum_\nu\frac{\nu!}{|\nu|!}\frac{|f_\nu|^2}{(|\nu|+1)^{\alpha+n}}\) and \(A_{\alpha,\#}^2:=\{f \text{ holomorphic on }{\mathbb B}^n:\|f\|_{\alpha,\#}-n-1\). Let \(T_\varphi^{(\alpha)}f=P_\alpha(\varphi f)\), \(\alpha>-1\), be the Toeplitz operator with symbol \(\varphi\in L^\infty({\mathbb B}^n)\), where \(P_\alpha:L^2({\mathbb B}^n,d\mu_\alpha)\to A_\alpha^2\) is the orthogonal projection. It may happen that \(T_\varphi^{(\alpha)}f\), for \(f\) in some dense subset, is given by an expression depending holomorphically on \(\alpha\), and extends by analyticity to the range \(\alpha>-n-1\). Some properties of such Toeplitz operators in the setting of the Bergman spaces \(A_\alpha^2\) of the unit ball \({\mathbb B}^n\) were studied by \textit{K. Chailuek} and \textit{B. C. Hall} [Integral Equations Oper. Theory 66, No. 1, 53--77 (2010; Zbl 1202.47029)]. The aim of the paper under review is to study the corresponding questions in the setting of smoothly bounded strictly pseudoconvex domains \(\Omega\) in \({\mathbb C}^n\). Assume that \(\varrho\in C^\infty(\overline{\Omega})\), \(\varrho>0\) on \(\Omega\), and \(\varrho=0\), \(\nabla\varrho\neq 0\) on \(\partial\Omega\). For \(\alpha<-1\), consider \(A_{\alpha,\varrho}^2(\Omega)=\{f\in L^2(\Omega,c_{\alpha,\varrho}\varrho^\alpha dz): f \text{ holomorphic on }\Omega\}\), where \(c_{\alpha,\varrho}=\left(\int_\Omega\varrho^\alpha dz\right)^{-1}\). Generalizing results by \textit{M. Vergne} and \textit{H. Rossi} [Acta Math. 136, 1--59 (1976; Zbl 0356.32020)], the authors define and study the corresponding family of spaces \(A_{\alpha,\#}^2(\Omega)\) for \(\alpha\in{\mathbb R}\). Toeplitz operators \(T_\varphi^{(\alpha,\varrho)}\), \(\varphi\in L^\infty(\Omega)\), defined on \(A_{\alpha,\varrho}^2(\Omega)\) then can also be ``analytically continued'' to \(A_{\alpha\,\#}^2(\Omega)\), \(\alpha\in{\mathbb R}\). The existence of such analytic continuation is established. Still further extensions to Sobolev spaces of holomorphic functions are also considered.