is a unital real algebra morphism.

Moreover, the following properties are true:

(1) for all $F \in \mathcal{O}_s(U, \mathcal{B}_n(V_n\mathbb{C}))$, $f \in \mathcal{O}_s(U)$, we have $(Ff)(\mathbf{T}) = F(\mathbf{T})f(\mathbf{T})$.

(2) for every polynomial $P(\zeta) = \sum_{j=0}^{m} A_j \zeta^j$, $\zeta \in \mathbb{C}$, with $A_j \in \mathcal{B}_n(V_n)$ for all $j = 0, 1, \dots, m$, we have $P(\mathbf{T}) = \sum_{j=0}^{m} A_j \mathbf{T}^j \in \mathcal{B}_n(V_n)$.

**Proof.** The proof follows the lines of that of Theorem 7 (see also Theorem 2 from [19]). We only note that a suitable version of the first part of Theorem 6 (or Theorem 1 and Remark 7 from [19]) is necessary, with minor modifications. The details are left to the reader.

**Remark 19.** As in [2], fixing an arbitrary family $\{T_j\}_{j=1}^n$, one considers the operator $\mathbf{T} = \sum_{j=1}^n T_j e_j$, which acts on $V_n$ via the formula

$$ \mathbf{T}(\mathbf{v}) = \sum_{j=1}^{n} \sum_{K} T_j(v_K)e_j e_K, \quad \mathbf{v} = \sum_{K} v_K e_K \in V_n. $$

In the previous statement we may replace the algebra $\mathcal{O}_s(U, \mathcal{B}_n(V_n\mathbb{C}))$ by the smaller algebra $\mathcal{O}_s(U, \mathbb{R}_n)$, regarding its elements as left multiplication operators (see Corollary 3). In this way, we obtain an analytic functional calculus for tuples of real operators, using analytic stem functions.

A parallel version of some results from this section was discussed in the first part of [2], using the slice regular functions to construct functional calculi.

The author is indebted to the referee for a careful reading of the original version of this paper, in particular for pointing out several typos.

**Conflict of interest:** Author states no conflict of interest.

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[19] F.-H. Vasilescu, Spectrum and Analytic Functional Calculus in Real and Quaternionic Frameworks, Pure and Applied Functional Analysis (to appear); see also arXiv:2003.05263.v2
[19] F.-H. Vasilescu, Spectrum and Analytic Functional Calculus in Real and Quaternionic Frameworks, Pure and Applied Functional Analysis (to appear); see also arXiv:2003.05263.v2