Norms of multiplication operators on Hardy spaces and weighted composition operators from Hardy spaces to weighted-type spaces on bounded symmetric domains
The author extends his former results about the norms of some operators on the Bergman and the Hardy space in the unit polydisk and the unit ball [Appl. Math. Comput. 215, No. 6, 2199--2205 (2009; Zbl 1186.32004)] to bounded symmetric domains \(D\). Let \(H(D)\) denote the ring of holomorphic functions on \(D\) and \[ H^p(D)= \Bigg\{f\in H(D)\mid \|f\|_{H^p(D)}:= \sup_{0\leq r<1} \bigg(\int_{\partial D}|f(r\zeta)|^p \,d\sigma(\zeta)\bigg)^{\frac{1}{p}}<\infty\Bigg\} \] the Hardy space of \(D\) for \(p>0\). Let \(u\in H(D)\) such that the multiplication operator \(M_u: H^p(D)\rightarrow H^p(D)\), \(M_uf(z):=u(z)f(z)\), is well-defined. The author calculates the norm of \(M_u\). \(M_u\) is bounded iff \(\|u\|_\infty=\sup_{z\in D}|u(z)|<\infty\), and then \(\|M_u\|=\|u\|_\infty\). More generally, let \(D=D_1\times\dots\times D_m\) be a decomposition of \(D\) with irreducible symmetric domains \(D_j\), \(1\leq j\leq m\). Let \(\varphi_j: D_j\rightarrow D_j\) be holomorphic, \(1\leq j\leq m\), \(\varphi=(\varphi_1,\dots,\varphi_m)\). Let \(\mu:D\rightarrow\mathbb R_{>0}\) be continuous and \(H^\infty_\mu(D):=\{f\in H(D)\mid \|f\|_{H^{\infty}_\mu(D)}=\sup\mu(z)|f(z)|<\infty\,\}\). Assume that, for given \(u\) and \(\mu\), the weighted composition operator \(uC_\varphi: H^p(D)\rightarrow H^\infty_\mu(D)\), \(uC_\varphi f(z):=u(z)f(\varphi(z))\), is well-defined and bounded. Then the norm of \(uC_\varphi\) can be precisely calculated in terms of \(\mu\), \(u\), \(\varphi\) and the Szegő-kernels \(S_j\) of \(D_j\), \(1\leq j\leq m\): \[ \|uC_\varphi\|=\sup_{z\in D}\mu(z)|u(z)|\prod_{j=1}^m \big(S_j(\varphi_j(z),\varphi_j(z))\big)^{\frac{1}{p}}. \]
scientific article; zbMATH DE number 5817377
scientific article