Toeplitzoperatører

Toeplitzoperatører are a class of linear operators that appear in various fields of mathematics and physics, particularly in signal processing and quantum mechanics. A Toeplitz operator is defined on a sequence space, such as l^2(Z), and its action on a sequence is determined by a specific function called the symbol. The defining characteristic of a Toeplitz operator is that its matrix representation with respect to the standard orthonormal basis is constant along each anti-diagonal. This means that if T is a Toeplitz operator, then the matrix element T_{i,j} depends only on the difference i-j. More formally, for a sequence (x_n)_{n in Z}, the Toeplitz operator T acting on it produces a sequence (y_n)_{n in Z} where y_n = sum_{k in Z} a_{n-k} x_k, and the sequence (a_k) is the generating sequence of the operator. This convolution structure is central to their properties. The algebra of Toeplitz operators is well-studied, and their spectral properties are closely related to the properties of their symbols. Compact Toeplitz operators are precisely those whose symbols vanish at infinity. They play a significant role in the study of discrete dynamical systems and the numerical solution of integral equations.