Toeplitz algebra and spectra of Toeplitz operators on the harmonic Dirichlet space

In this paper, we study the algebraic and spectral properties of Toeplitz operators on the harmonic Dirichlet space. We completely characterize the Toeplitz algebra and Hankel algebra on such function space. Moreover, using some techniques in algebraic curves theory, we investigate the spectra of Toeplitz operators with analytic polynomial symbols and show that the spectrum of the Toeplitz operator Tp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_p$$\end{document} with p a polynomial of degree less than 4 is equal to the union of a closed curve and at most finitely many isolated points.