The Bernstein operator maps $f\in C[0,1]$ to its Bernstein polynomial $B_n f.$ The eigenvalues and eigenfunctions of the Bernstein operator on $C[0,1]$ have been described in [1]. Similar description has been obtained for the $q$-Bernstein polynomials in [2]. The study of $q$-Bernstein polynomials in the case $0<q<1$ leads to the following definition.

**Definition**. Let $0<q< 1.$ The *limit q-Bernstein
operator* on $C[0,1]$ is given by:
$$ B_{\infty,q}:f\mapsto B_{\infty,q}f,
$$where
$$(B_{\infty,q}f)(x) = \left\{
\begin{array}{ll} \displaystyle \prod_{j=0}^{\infty}(1-q^jx)\cdot \sum_{k=0}^{\infty}\frac{f(1-q^k)\,x^k}{(1-q)\dots
(1-q^k)},
& x\in [0,1),\\
f(1), & x=1.
\end{array}
\right.
$$

**Problem**. Find all $f\in C[0,1]$ so that
$$B_{\infty,q}f=\lambda f,\;\;\lambda \in {\bf C}\setminus \{0\}.$$

Conjecture: If $B_{\infty,q}f=\lambda f,\;\lambda \neq 0,$ then $f$ is a polynomial and $\lambda\in \{q^{m(m-1)/2}\}_{m=0}^{\infty}.$

Remark. The conjecture has been proved under some additional conditions on the smoothness of $f$ at 1 (for example, for $f\in {\rm Lip}\,\alpha$) in [3], Corollary 5.6.

[1] S. Cooper, S. Waldron, The Eigenstructure of
the Bernstein Operator, *J. Approx. Theory*, **105**, 2000,
133-165.

[2] S. Ostrovska, M. Turan, On the eigenvectors of the q-Bernstein operators, *Mathematical Methods in the Applied Sciences*, Vol **37**, Issue 4 (2014), pp. 562-570.

[3] S. Ostrovska, On the improvement of analytic
properties under the limit $q$-Bernstein
operator, *J. Approx. Theory*, **138**, 2006, 37-53.