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Higher order Hankel forms. (English) Zbl 0944.47019
Curto, Raúl E. (ed.) et al., Multivariable operator theory. A joint summer research conference on multivariable operator theory, July 10-18, 1993, University of Washington, Seattle, WA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 185, 283-306 (1995).
Let $$\mathbb{D}$$ denote the open unit disk in the complex plane $$\mathbb{C}$$ and $$L^2$$ the usual space of square integral measurable functions on $$\mathbb{D}$$ with area measure. Let $$H$$ be the space of $$L^2$$ consisting of analytic functions. The small Hankel operator on $$H$$ with symbol $$b$$ (where $$b$$ is a measurable function on $$\mathbb{D}$$) is usually defined by the equivalent of (1) $$H_bf= Q(bf)$$, where $$Q$$ is the orthogonal projection from $$L^2$$ to $$\overline H$$, the space of complex conjugates of analytic functions in $$L^2$$. Of course $$b$$ cannot be completely arbitrary for (1) to be defined, but it is known that it need not be bounded to make sense of $$H_b$$. Clearly, $$H_b$$ is determined by the bilinear form (2) $$B(f,g)= \int_{\mathbb{D}}bfg dA$$. Any bilinear form $$B$$ on $$H$$ may be viewed as a linear functional $$\widetilde B$$ on the tensor product $$H\otimes H$$. Since the space $$H\otimes H$$ can be identified with linear combinations of functions of the form $$f(z)g(w)$$, $$f,g\in H$$, on $$\mathbb{D}\times \mathbb{D}$$, the effect of (2) is that $$\widetilde B(h)$$ depends only on the values of $$h$$ on the diagonal $$\Delta= \{(z,z): z\in\mathbb{D}\}$$.
The set of Hilbert-Schmidt bilinear forms is a Hilbert space and, being identified with linear functionals on $$H\otimes H$$, can be identified with the Hilbert space completion of $$H\otimes H$$ (which we will denote by the same symbol $$H\otimes H$$). In this context, a Hankel operator is identified with an element of $$V_\Delta^\perp$$, where $$V_\Delta= \{f\in H\otimes H: f|_\Delta\equiv 0\}$$.
Now $$H$$ is actually a Hilbert module over $$H^\infty$$, the algebra of bounded analytic functions on $$\mathbb{D}$$. From (2) we see that $$\delta_a(B)\equiv B(af,g)- B(f, ag)= 0$$ for all $$f,g\in H$$ and $$a\in H^\infty$$.
This paper is concerned with generalizations of Hankel operators. In the Bergman space context, a Hankel form (of type $$0$$) can be defined in any of three ways:
(a) $$\widetilde B(h)$$ depends only on the values of $$h(z,w)$$ on $$\Delta$$;
(b) $$\widetilde B\in V_\Delta^\perp$$;
(c) $$\delta_a(B)= 0$$.
Each of these gives rise to a method of defining Hankel forms of higher-order. A Hankel form of order $$s$$ is a bilinear form $$B$$ on $$H$$ such that:
$$(\text{a}')$$ $$\widetilde B(h)$$ depends only on the values of $$h(z,w)$$ on $$\Delta$$ and its derivatives perpendicular to $$\Delta$$ of order at most $$s$$;
$$(\text{b}')$$ $$\widetilde B\perp(V_\Delta)^{s+ 1}$$, where $$(V_\Delta)^s$$ denotes limits of sums of products of $$s$$ functions in $$V_\Delta$$ (or rather in a dense subset of $$V_\Delta$$ so that these products belong to $$H\otimes H$$);
$$(\text{c}')$$ $$(\delta_a)^s(B)= 0$$.
This paper shows that these different definitions are equivalent in some of the standard settings ($$H$$ is a Bergman space or a Hardy space), surveys some of the known results about higher-order Hankel operators from this point of view (work on higher-order Hankel operators originally arose out of an examination of the action of Möbius transformations of the disk on $$L^2$$), and raises several questions about the state of affairs when $$H$$ is a more general Hilbert space of functions with a reproducing kernel.
For the entire collection see [Zbl 0819.00022].

##### MSC:
 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 46E15 Banach spaces of continuous, differentiable or analytic functions 30D99 Entire and meromorphic functions of one complex variable, and related topics