Hankel operators over the complex Wiener space (Q1424813)

The author investigates Hankel forms and small Hankel operators on the Hilbert space \({\mathcal H}L^2(\mu_c)\) of holomorphic square integrable Wiener functionals, that is, the \(L^2(\mu_c)\)-closure of \({\mathcal P}(Z)\), the algebra of holomorphic polynomials generated by the complex Gaussian variables. Here \(\mu_c\) is the Wiener measure with variance parameter \(c\) and \(W_{\mathbb C}\) is the complex Wiener space over some fixed interval \([0,T]\). Theorem 3.2. Let \(e^{-tN}\) be the Ornstein--Uhlenbeck semigroup on \({\mathcal H}L^2(\mu_c)\) and let \(\varphi\in{\mathcal H}L^p(\mu_c)\) with \(p\geq 2\). Set \(b:=e^{-tN}\varphi\) with \(t\geq (c/2)\log p'\), where \(1/p+1/p'=1\), and define \(\Gamma_b(f,g):=\int_{W_{\mathbb C}} \bar{b}fg\,d\mu_c\) on \({\mathcal P}(Z)\). Then \(\Gamma_b\) extends by continuity to \({\mathcal H}L^2(\mu_c)\): for all \(f,g\in{\mathcal P}(Z_n,n\in{\mathbb N})\), \[ \left| \int_{W_{\mathbb C}}\bar{b}fg\,d\mu_c\right| \leq \| \varphi\| _{L^p(\mu_c)} \| f\| _{L^2(\mu_c)} \| g\| _{L^2(\mu_c)}. \] The latter result gives conditions for the boundedness of the small Hankel operator \(H_b\) satisfying \(\Gamma_b(f,g)=\langle g,H_bf\rangle\) for all \(f,g\in{\mathcal H}L^2(\mu_c)\). Further, the author gives criteria under which \(H_b\) is Hilbert--Schmidt. Theorem 4.2. Let \(b\in{\mathcal H}L^2(\mu_c)\). Then \(H_b\) is Hilbert--Schmidt on \({\mathcal H}L^2(\mu_c)\) if and only if \(b\in {\mathcal H}L^2(\mu_{2c})\). Moreover, in the case that \(H_b\) is Hilbert--Schmidt, the Hilbert--Schmidt norm of \(H_b\) is equal to the norm of \(b\) in \(L^2(\mu_{2c})\). Finally, a representation of \(H_b\) by an integral operator is given. The proofs use the hypercontractivity of the Ornstein--Uhlenbeck semigroup and approximation by finitely many variables.