Infinite Toeplitz Matrices

Abstract

Given a sequence $\left\{ {{a_n}} \right\}_{n = - \infty }^\infty$ of complex numbers, a _n ∈ C, when does the matrix

$$A = \left( {\begin{array}{*{20}{c}} {{a_0}} & {{a_{ - 1}}} & {{a_{ - 2}}} & \cdots \\ {{a_1}} & {{a_0}} & {{a_{ - 1}}} & \cdots \\ {{a_2}} & {{a_1}} & {{a_0}} & \cdots \\ \cdots & \cdots & \cdots & \cdots \\ \end{array} } \right)$$

((1.1))

induce a bounded operator on l² ≔ l²(Z₊), where Z₊ is the set of nonnegative integers, Z₊ ≔ {0,1,2,…}? The answer is classical result by Otto Toeplitz.