An application of averaging operators to multilinearity
For compact metric spaces \(K\), \(S\) and a continuous surjection \(\varphi: K\to S\), let the linear isometry \(\varphi^ 0: C(S)\to C(K)\) be given by \(\varphi^ 0 f= f\circ\varphi\). \(\varphi\) is said to admit a (regular) averaging operator if there exists a (contractive) projection from \(C(K)\) onto \(\varphi^ 0(C(S))\). As an application of the existence (due to A. Pełczyński) of averaging operators, the author proves that certain multilinear operators between spaces \(C(K)\) of continuous functions can be represented as products of bounded linear operators and the (pointwise) multiplication operator on \(C(K)\). More precisely (and more generally), let \(K\) and \(S\) be compact spaces, \(K\) metric uncountable, let \(P\) be a continuous symmetric \(M\)-linear map \(C(K)\times\cdots\times C(K)\to C(S)\), \(M>1\). Assume that, for each \(t\in S\), the map \[ f_ 1(x_ 1)\cdots f_ M(x_ M)\to P(f_ 1,\dots,f_ M)(t),\quad\text{where } f_ j\in C(K), \] can be extended to a continuous linear form \(m^{(t)}\) on \(C(K^ M)\) which depends continuously on \(t\) when \(C(K^ M)^*\) is endowed with the \(\text{weak}^*\)-topology. Then there exist continuous linear operators \(A\in L(C(K),C(K))\), \(B\in L(C(K),C(S))\) such that \[ P(f_ 1,\dots,f_ M)= B\left(\prod^ M_{j=1} Af_ j\right)\qquad\text{for all }f_ 1,\dots,f_ M\in C(K). \] Conversely, if \(P\) can be represented in this form, then the assumptions on \(m^{(t)}\) hold. -- There is a similar result for continuous symmetric \(M\)-linear maps from \(C(K)\times\cdots \times C(K)\) into a Banach space \(X\). The author also discusses the case when \(K\) is non-metrizable.
products of bounded linear operators and the pointwise multiplication operator
scientific article; zbMATH DE number 537331