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Invariant subspaces, quasi-invariant subspaces, and Hankel operators. (English) Zbl 1045.47022
The authors study algebraic properties of small Hankel operators on Bergman spaces of bounded symmetric domains \(\Omega\subset \mathbb{C}^n\). Here, the Bergman space \(L^2_a(\Omega)\) is the closed subspace of \(L^2(\Omega)\) consisting of analytic functions and the small Hankel operator \(\Gamma_\varphi\) with symbol \(\varphi\in L^2(\Omega)\) is the operator densely defined on \(L_a^2\) by \(\Gamma_{\varphi}f=P(\varphi \widehat f)\), where \(\widehat f(z)=f(\overline z)\) and \(P\) is the orthogonal projection from \(L^2\) onto \(L^2_a\). First, some algebraic properties of small Hankel operators are obtained using explicit expressions of reproducing kernels. Next, the structure of \(M^{\perp}\) for finite-codimensional invariant subspaces \(M\subset L^2(\Omega)\) (i.e., \(pM\subset M\) for all polynomials \(p\)) is described and used to obtain a complete characterization of small Hankel operators of finite rank. A closed subspace \(M\) of the Segal-Bargmann space \(L^2_a(\mathbb{C}^n)\) is called quasi-invariant if \(pM\cap L^2_a(\mathbb{C}^n)\subset M\) for all polynomials \(p\). In the last two sections of this paper, conditions under which two quasi-invariant subspaces are similar (unitarily equivalent) are proved and used to describe finite-rank small Hankel operators on \(L^2_a(\mathbb{C}^n)\).

MSC:
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47A15 Invariant subspaces of linear operators
32A15 Entire functions of several complex variables
32A36 Bergman spaces of functions in several complex variables
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