关键词: Toeplitz matrix, infinite matrix, block matrix, exponential, functional calculus

Abstract: The matrix Wiener algebra, $\mathcal{W}_N:=\mathrm{M}_{N} (\mathcal{W})$ of order $N>0$, is the matrix algebra formed by $N \times N$ matrices whose entries belong to the classical Wiener algebra $\mathcal{W}$ of functions with absolutely convergent Fourier series. A block-Toeplitz matrix $T(a)=[A_{i,j}]_{i,j \geq 0}$ is a block semi-infinite matrix such that its blocks $A_{i,j}$ are finite matrices of order $N$, $A_{i,j}=A_{r,s}$ whenever $i-j=r-s$ and its entries are the coefficients of the Fourier expansion of the generator $a:\mathbb{T} \rightarrow \mathrm{M}_{N} (\mathbb{C})$. Such a matrix can be regarded as a bounded linear operator acting on the direct sum of $N$ copies of $L^2(\mathbb{T})$. We show that $\exp(T(a))$ differes from $T(\exp(a))$ only in a compact operator with a known bound on its norm. In fact, we prove a slightly more general result: for every entire function $f$ and for every compact operator $E$, there exists a compact operator $F$ such that $f(T(a)+E)=T(f(a))+F$. We call these $T(a)+E's$ matrices, the quasi block-Toeplitz matrices, and we show that via a computation-friendly norm, they form a Banach algebra. Our results generalize and are motivated by some recent results of Dario Andrea Bini, Stefano Massei and Beatrice Meini.

Key words: Toeplitz matrix, infinite matrix, block matrix, exponential, functional calculus