Let X\documentclass[12pt]{minimal}
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\begin{document}$\mathcal {X}$\end{document} be a separable Hilbert space with norm ⋅\documentclass[12pt]{minimal}
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\begin{document}$\left \|{\cdot }\right \|$\end{document} and let T > 0. Let Q be a linear, self-adjoint, positive, trace class operator on X\documentclass[12pt]{minimal}
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\begin{document}$\mathcal {X}$\end{document}, let F:X→X\documentclass[12pt]{minimal}
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\begin{document}$F:\mathcal {X}\rightarrow \mathcal {X}$\end{document} be a (smooth enough) function and let W(t) be a X\documentclass[12pt]{minimal}
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\begin{document}$\mathcal {X}$\end{document}-valued cylindrical Wiener process. For α ∈ [0, 1/2] we consider the operator A:=−(1/2)Q2α−1:Q1−2α(X)⊆X→X\documentclass[12pt]{minimal}
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\begin{document}$A:=-(1/2)Q^{2\alpha -1}:Q^{1-2\alpha }(\mathcal {X})\subseteq \mathcal {X}\rightarrow \mathcal {X}$\end{document}. We are interested in the mild solution X(t, x) of the semilinear stochastic partial differential equation
dX(t,x)=AX(t,x)+F(X(t,x))dt+QαdW(t),t∈(0,T];X(0,x)=x∈X,\documentclass[12pt]{minimal}
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\begin{document}$$ \left\{\begin{array}{ll} dX(t,x)=\left( AX(t,x)+F(X(t,x))\right)dt+ Q^{\alpha}dW(t), & t\in(0,T];\\ X(0,x)=x\in \mathcal{X}, \end{array}\right. $$\end{document}
and in its associated transition semigroup
P(t)φ(x):=E[φ(X(t,x))],φ∈Bb(X),t∈[0,T],x∈X;\documentclass[12pt]{minimal}
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\begin{document}$$ P(t)\varphi(x):=\mathbb{E}[\varphi(X(t,x))], \qquad \varphi\in B_{b}(\mathcal{X}),\ t\in[0,T],\ x\in \mathcal{X}; $$\end{document}
where Bb(X)\documentclass[12pt]{minimal}
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\begin{document}$B_{b}(\mathcal {X})$\end{document} is the space of the bounded and Borel measurable functions. We will show that under suitable hypotheses on Q and F, P(t) enjoys regularizing properties, along a continuously embedded subspace of X\documentclass[12pt]{minimal}
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\begin{document}$\mathcal {X}$\end{document}. More precisely there exists K := K(F, T) > 0 such that for every φ∈Bb(X)\documentclass[12pt]{minimal}
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\begin{document}$\varphi \in B_{b}(\mathcal {X})$\end{document}, x∈X\documentclass[12pt]{minimal}
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\begin{document}$x\in \mathcal {X}$\end{document}, t ∈ (0, T] and h∈Qα(X)\documentclass[12pt]{minimal}
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\begin{document}$h\in Q^{\alpha }(\mathcal {X})$\end{document} it holds
|P(t)φ(x+h)−P(t)φ(x)|≤Kt−1/2∥Q−αh∥.\documentclass[12pt]{minimal}
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\begin{document}$$ |P(t)\varphi(x+h)-P(t)\varphi(x)|\leq Kt^{-1/2}\|Q^{-\alpha}h\|. $$\end{document}
