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].

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].

Reviewer: Daniel H.Luecking (MR 96g:47020)

##### 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 |