Let \(T=(T_{1},\ldots , T_{n})\) be a commuting n-tuple of operators on a complex Hilbert space H and let \(\left( \begin{array}{c} T_{1} \\ \vdots \\ T_{n} \end{array} \right) =\left( \begin{array}{c} V_{1} \\ \vdots \\ V_{n} \end{array} \right) P=\left( \begin{array}{c} V_{1}P \\ \vdots \\ V_{n}P \end{array} \right) \) be the polar decomposition of the column operator \(\left( \begin{array}{c} T_{1} \\ \vdots \\ T_{n} \end{array} \right) \) , where \(P=\sqrt{\sum \limits _{i=1}^{n}T_{i}^{*}T_{i}}\) \(\left( \begin{array}{c} V_{1} \\ \vdots \\ V_{n} \end{array} \right) \) is the partial isometry, such that \(\bigcap \nolimits _{i=1}^{n} \textrm{ker} ({\textrm{T}}_{\textrm{i}})=\bigcap \nolimits _{i=1}^{n} \textrm{ker} (\textrm{V}_{\textrm{i}})=\textrm{ker} (\textrm{P})\) . Let f and g be two continuous functions from \(\mathbb {R}^{+}\) to \(\mathbb {R}^{+}\) , such that \(f(x)g(x)=x\) , for every x in \(\mathbb {R}^{+}\) and \(g(0)=0\) . The generalized spherical (f, g)-Aluthge transform of T is defined by the n-tuple (necessarily commuting) (see [22]) \(\begin{aligned} \Delta _{f, g}(T)=(f(P)V_{1}g(P), f(P)V_{2}g(P), \ldots , f(P)V_{n}g(P) ). \end{aligned}\) Let \(k\in \{0, \ldots , n\}\) , we denote by \(\sigma _{e}^{\pi , k},\) \(\sigma _{e}^{\delta , k},\) \(\sigma _{\delta , k}\) and \(\sigma _{\pi , k}\) the Slodkowski’s spectral systems. In this paper, we show that for every \( k\in \{0, \ldots , n\}\) , \(\sigma _{\pi , k}(T)=\sigma _{\pi , k}(\Delta _{f, g}(T)),\) \(\sigma _{\delta , k}(T)\subseteq \sigma _{\delta , k}(\Delta _{f, g}(T))\subseteq \sigma _{\delta , k}(T)\cup \{0\},\) \(\sigma _{e}^{\pi , k}(T)\backslash \{0\}=\sigma _{e}^{\pi , k}(\Delta _{f, g}(T))\backslash \{0\}\) and \(\sigma _{e}^{\delta , k}(T)\backslash \{0\}=\sigma _{e}^{\delta , k}(\Delta _{f, g}(T))\backslash \{0\}.\) In particular, we obtain that \(\sigma _{T}(T)=\sigma _{T}(\Delta _{f, g}(T))\) and \(\sigma _{T_{e}}(T)=\sigma _{T_{e}}(\Delta _{f, g}(T))\) , where \(\sigma _{T}\) and \(\sigma _{T_{e}}\) denote the Taylor spectrum and the essential Taylor spectrum respectively. Moreover if \(f(0)\ne 0\) , then \(\sigma ^{'}(\Delta _{f, g}(T))=\sigma ^{'}(T)\) , \(\forall \sigma ^{'}\in \{\sigma _{\pi , k}, \sigma _{\delta , k}, \sigma ^{\pi , k}_{e}, \sigma ^{\delta , k}_{e}, k\in \{0, \ldots , n\}\}\) .
