article published Jan Maas author 4C5696CE-F248-11E8-B48F-1D18A9856A870000-0002-0845-1338 Jan Van Neerven author Let (E, H, μ) be an abstract Wiener space and let DV : = V D, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space under(H, {combining low line}). Given a bounded operator B on under(H, {combining low line}), coercive on the range over(R (V), -), we consider the operators A : = V* B V in H and under(A, {combining low line}) : = V V* B in under(H, {combining low line}), as well as the realisations of the operators L : = DV* B DV and under(L, {combining low line}) : = DV DV* B in Lp (E, μ) and Lp (E, μ ; under(H, {combining low line})) respectively, where 1 &lt; p &lt; ∞. Our main result asserts that the following four assertions are equivalent: (1)D (sqrt(L)) = D (DV) with {norm of matrix} sqrt(L) f {norm of matrix}p {minus tilde} {norm of matrix} DV f {norm of matrix}p for f ∈ D (sqrt(L));(2)under(L, {combining low line}) admits a bounded H∞-functional calculus on over(R (DV), -);(3)D (sqrt(A)) = D (V) with {norm of matrix} sqrt(A) h {norm of matrix} {minus tilde} {norm of matrix} V h {norm of matrix} for h ∈ D (sqrt(A));(4)under(A, {combining low line}) admits a bounded H∞-functional calculus on over(R (V), -). Moreover, if these conditions are satisfied, then D (L) = D (DV2) ∩ D (DA). The equivalence (1)-(4) is a non-symmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where under(H, {combining low line}) = H, V = I, B = frac(1, 2) I). A one-sided version of (1)-(4), giving Lp-boundedness of the Riesz transform DV / sqrt(L) in terms of a square function estimate, is also obtained. As an application let -A generate an analytic C0-contraction semigroup on a Hilbert space H and let -L be the Lp-realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A. The boundedness of the Riesz transform is used to obtain an Lp-domain characterisation for the operator L. Academic Press2009 10.1016/j.jfa.2009.07.001 25782410 - 2475 yes Maas J, Van Neerven J. 2009. Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces. Journal of Functional Analysis. 257(8), 2410–2475. Maas, J., &#38; Van Neerven, J. (2009). Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces. <i>Journal of Functional Analysis</i>. Academic Press. <a href="https://doi.org/10.1016/j.jfa.2009.07.001">https://doi.org/10.1016/j.jfa.2009.07.001</a> J. Maas, J. Van Neerven, Journal of Functional Analysis 257 (2009) 2410–2475. Maas, Jan, and Jan Van Neerven. “Boundedness of Riesz Transforms for Elliptic Operators on Abstract Wiener Spaces.” <i>Journal of Functional Analysis</i>, vol. 257, no. 8, Academic Press, 2009, pp. 2410–75, doi:<a href="https://doi.org/10.1016/j.jfa.2009.07.001">10.1016/j.jfa.2009.07.001</a>. Maas, Jan, and Jan Van Neerven. “Boundedness of Riesz Transforms for Elliptic Operators on Abstract Wiener Spaces.” <i>Journal of Functional Analysis</i>. Academic Press, 2009. <a href="https://doi.org/10.1016/j.jfa.2009.07.001">https://doi.org/10.1016/j.jfa.2009.07.001</a>. J. Maas and J. Van Neerven, “Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces,” <i>Journal of Functional Analysis</i>, vol. 257, no. 8. Academic Press, pp. 2410–2475, 2009. Maas J, Van Neerven J. Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces. <i>Journal of Functional Analysis</i>. 2009;257(8):2410-2475. doi:<a href="https://doi.org/10.1016/j.jfa.2009.07.001">10.1016/j.jfa.2009.07.001</a> 21192018-12-11T11:55:49Z2021-01-12T06:55:25Z