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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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