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On Toeplitz-invariant subspaces of the Bergman space. (English) Zbl 0772.30042

Let \(A_ 2\) be the Bergman space of all those analytic functions in the open unit disk for which \[ \| f\|^ 2_{A_ 2}:={1\over \pi}\iint_{\mathbb{D}} | f(z)|^ 2dA(z)<\infty, \] where \(dA\) denotes 2-dimensional Lebesgue measure. As usual, \(k(z,w)={1\over\pi}\cdot{1\over(1-z\bar w)^ 2}\) is the associated Bergman kernel. The main result of this interesting paper is a complete description of those finite-dimensional subspaces \(M\subseteq H^ \infty\subseteq A_ 2\), \(1\in M\), which are invariant under the quadratic Toeplitz operator \(T\), i.e., for which \(TM\subseteq M\). Recall that \[ T(z)=T_ g(z)={1\over \pi}\iint_{\mathbb{D}} | g(z)|^ 2 k(z,w)dA(w),\quad g\in A_ 2. \] It depends on an extension of the notion of inner functions based on H. Hedenmalm’s fundamental work [J. Reine Angew. Math. 422, 45-68 (1991; Zbl 0734.30040)]. In fact, for every subspace \(L\subseteq A_ 2\) a function \(G\in A_ 2\) is said to be an \(L\)-inner function if
i) \(\| G\|_{A_ 2}=1\),
ii) \(gG\in A_ 2\) for every \(g\in L\),
iii) \(\iint_{\mathbb{D}}(| G|^ 2-1)g dA=0\) for all \(g\in L\).
If \(L=H^ \infty\), then it was proved by Hedenmalm [loc. cit.] that these inner functions are exactly those functions which are norm- expansive \(L\)-multipliers for \(A_ 2\), i.e., which satisfy i) and
iv) \(\| Gg\|_{A_ 2}\geq\| g\|_{A_ 2}\) for every \(g\in L\).
The authors now show that a finite-dimensional subspace \(M\subseteq H^ \infty\subseteq A_ 2\), \(1\in M\), is \(T\)-invariant if and only if every \(M\)-inner function is a norm-expansive \(M\)-multiplier for \(A_ 2\).
It is also shown that, whenever \(M\) is a finite-dimensional subspace of polynomials \((1\in M)\), then \(M\) is \(T\)-invariant if and only if \[ M=\text{span}\{z^{kj}: j=0,1,\dots,n\} \] for integers \(k>0\) and \(n\geq 0\).
Analogous results for the standard Hardy space \(H^ 2\) are obtained.

MSC:

30H05 Spaces of bounded analytic functions of one complex variable
46E20 Hilbert spaces of continuous, differentiable or analytic functions
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators

Citations:

Zbl 0734.30040
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