Completion of space of finite rank quadratic operators (Q1312274)

If \(H\) and \(G\) are Hilbert spaces over the same scalar field, an operator \(Q: H\to G\) is called quadratic if there is a bilinear operator \(\widehat Q\) from \(H\times H\) to \(G\) so that \(\widehat Q(x,x)= Q(x)\) for all \(x\) in \(H\). This paper continues the study of such operators begun in earlier work [\textit{J. C. Amson} and the author, Bull. Aust. Math. Soc. 41, No. 1, 123-134 (1990; Zbl 0678.46045)] by focusing on repesentations of finite rank quadratic operators and vectors spaces of such operators. In particular, it is shown that the closure in so-called ``Hilbert-Schmidt quadratic norm'' (whose definition here is given in terms of a sequence \(\{L^ k\}\), which is never defined) of the space of finite-dimensional quadratic operators is the space of all Hilbert-Schmidt quadratic operators on \(H\).