arXiv 2102.043022021-02-08math.NAcs.NAmath.OA

拟Toeplitz矩阵的几何平均

Geometric means of quasi-Toeplitz matrices

Dario A. Bini, Bruno Iannazzo, Jie Meng

备注：: 33 pages, 4 figures

摘要：

我们研究几何型的拟Toeplitz矩阵，即半无限矩阵$A=(a_{i,j})_{i,j=1,2,\ldots}$的形式$A=T(a)+E$，其中$E$表示一个紧凑的运营商，和$T(a)$是一个半无限Toeplitz矩阵与功能$a$，傅立叶级数$\sum_{\ell=-\infty}^{\infty} a_\ell e^{\mathfrak i \ell t}$，在这个意义上，$(T(a))_{i,j}=a_{j-i}$。如果$a$是\rv \且本质有界，则这些矩阵表示$\ell^2$上的有界自伴算子。我们考虑的情况下，$a$是一个连续的功能，其中拟Toeplitz矩阵符合一个经典的Toeplitz代数，和$a$的情况下，在维纳代数，也就是说，有绝对收敛的傅立叶级数。证明了若$a_1,\ldots,a_p$是连续正函数，或满足一定条件的Wiener代数，则与$a_1,\ldots,a_p$相关的拟Toeplitz正定矩阵的几何型均值，如ALM、NBMP和Karcher均值，都是与几何均值$(a_1\cdots a_p)^{1/p}$相关的拟Toeplitz矩阵，它们的区别仅在于紧校正.我们通过数值试验表明，这些运营商的手段，实际上可以近似。

英文摘要：

We study means of geometric type of quasi-Toeplitz matrices, that are semi-infinite matrices $A=(a_{i,j})_{i,j=1,2,\ldots}$ of the form $A=T(a)+E$, where $E$ represents a compact operator, and $T(a)$ is a semi-infinite Toeplitz matrix associated with the function $a$, with Fourier series $\sum_{\ell=-\infty}^{\infty} a_\ell e^{\mathfrak i \ell t}$, in the sense that $(T(a))_{i,j}=a_{j-i}$. If $a$ is \rv\ and essentially bounded, then these matrices represent bounded self-adjoint operators on $\ell^2$. We consider the case where $a$ is a continuous function, where quasi-Toeplitz matrices coincide with a classical Toeplitz algebra, and the case where $a$ is in the Wiener algebra, that is, has absolutely convergent Fourier series. We prove that if $a_1,\ldots,a_p$ are continuous and positive functions, or are in the Wiener algebra with some further conditions, then means of geometric type, such as the ALM, the NBMP and the Karcher mean of quasi-Toeplitz positive definite matrices associated with $a_1,\ldots,a_p$, are quasi-Toeplitz matrices associated with the geometric mean $(a_1\cdots a_p)^{1/p}$, which differ only by the compact correction. We show by numerical tests that these operator means can be practically approximated.

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