讲座内容：

In this paper, we investigate a class of Hardy-type operators generated by dilation averages on the Hardy space over the upper half-plane. We first prove that Hardy-type operator H is bounded and obtain the exact value of its operator norm. By exploiting an orthonormal basis adapted to the underlying dilation structure, we show that I − H admits a bilateral shift model. Based on this structural analysis, we establish that the point spectrum of H is empty and give a recise characterization of its spectrum as the circle. Finally, using the boundary operator of H as a bridge, we also obtain an explicit solution operator for a class of piecewise Euler-type first-order differential equations with switching interface x = 0, thereby relating the well-posedness of piecewise dynamical systems directly to (I − Hb) −1 。