In this introductory review, we study Hankel and Toeplitz opera-
tors considering them as acting on certain spaces of analytic functions, namely
Hardy spaces and compare their spectral properties such as their compactness
criteria. In contrast to Toeplitz operators, the symbol of a Hankel operator is
not uniquely determined by the operator. We also connect Toeplitz operators
with Fredholm operators and give some of the most beautiful properties of
Toeplitz operators such as the essential spectrum of Toeplitz operator with
continuous symbol and the index of Toeplitz operator introducing Fredholm
operators firstly.