In this paper, first we consider cross-diagonal, in particular off-diagonal, operator matrices; those operator matrices such that all entries except those on the main and off diagonals are zero. We show that this operator matrix is unitarily equivalent to a block diagonal operator matrix whose diagonal blocks are all two-by-two, except at most one of them which is one-by-one. Then, using this unitary equivalence, we show that any left circulant operator matrix is unitarily equivalent to a direct sum of a one-by-one operator and an off-diagonal operator matrix. As an application, we give equalities for the numerical radius of some important operator matrices. In particular, for the following left circulant operator matrix, we show that w⎛⎝⎜⎜⎜⎜⎜⎡⎣⎢⎢⎢⎢⎢A1A2⋮AnA2…⋱A1…An⋱…AnA1⋮An−1⎤⎦⎥⎥⎥⎥⎥⎞⎠⎟⎟⎟⎟⎟=max1≤k≤nw⎛⎝⎜⎜⎜⎜⎜⎡⎣⎢⎢⎢⎢⎢0∑m=1nω(n−k)(m−1)Am∑m=1nωk(m−1)Am0⎤⎦⎥⎥⎥⎥⎥⎞⎠⎟⎟⎟⎟⎟,$\begin{array}{rl}& w\left(\left[\begin{array}{cccc}{A}_1& A_2& \dots & A_n\\ A_2& \dots & A_n& A_1\\ ⋮& \ddots & \ddots & ⋮\\ A_n& A_1& \dots & A_{n-1}\end{array}\right]\right)\\ & =\underset{1\le k\le n}{max}w\left(\left[\begin{array}{cc}0& {\displaystyle \sum _{m=1}^n{\omega }^{k(m-1)}{A}_m}\\ {\displaystyle \sum _{m=1}^n{\omega }^{(n-k)(m-1)}{A}_m}& 0\end{array}\right]\right),\end{array}$$\begin{array}{rl}& w\left(\left[\begin{array}{cccc}{A}_1& A_2& \dots & A_n\\ A_2& \dots & A_n& A_1\\ ⋮& \ddots & \ddots & ⋮\\ A_n& A_1& \dots & A_{n-1}\end{array}\right]\right)\\ & =\underset{1\le k\le n}{max}w\left(\left[\begin{array}{cc}0& {\displaystyle \sum _{m=1}^n{\omega }^{k(m-1)}{A}_m}\\ {\displaystyle \sum _{m=1}^n{\omega }^{(n-k)(m-1)}{A}_m}& 0\end{array}\right]\right),\end{array}$where ω=e2π in$\omega ={\mathrm{e}}^{\frac{2\pi \text{ i}}{n}}$$\omega ={\mathrm{e}}^{\frac{2\pi \text{ i}}{n}}$ is the nth root of unity. Meanwhile, some inequalities for general operator matrices are obtained.

在本文中，首先我们考虑跨对角，特别是非对角算子矩阵；这些运算符矩阵使得除主对角线上和非对角线上的条目之外的所有条目均为零。我们证明该算子矩阵酉等价于块对角算子矩阵，其对角线块都是二乘二的，除了最多一个是一对一的。然后，使用这种酉等价，我们证明任何左循环算子矩阵都酉等价于一对一算子和非对角算子矩阵的直和。作为一个应用，我们给出了一些重要算子矩阵的数值半径的等式。特别是，对于以下左循环算子矩阵，我们证明w⎛⎝⎜⎜⎜⎜⎜⎡⎣⎢⎢⎢⎢⎢A1A2⋮AnA2……⋱A1……An⋱……AnA1⋮An − 1⎤⎦⎥⎥⎥⎥⎥⎞⎠⎟⎟⎟⎟⎟=最大限度1≤k≤n _ _ _ _w⎛⎝⎜⎜⎜⎜⎜⎡⎣⎢⎢⎢⎢⎢0Σ米= 1nω( n − k ) ( m − 1 )A米Σ米= 1nωk ( m − 1 )A米0⎤⎦⎥⎥⎥⎥⎥⎞⎠⎟⎟⎟⎟⎟,$\begin{array}{rl}& w（\left[\begin{array}{cccc}{A}_1& A_2& \dots \dots & A_n\\ A_2& \dots \dots & A_n& A_1\\ ⋮& \ddots & \ddots & ⋮\\ A_n& A_1& \dots \dots & A_{n-1}\end{array}\right]）\\ & =\underset{1\le k\le n}{最大限度}w（\left[\begin{array}{cc}0& {\displaystyle \underset{米=1}{\overset{n}{\Sigma }}{\omega }^{k（米-1）}{A}_{米}}\\ {\displaystyle \underset{米=1}{\overset{n}{\Sigma }}{\omega }^{（n-k）（米-1）}{A}_{米}}& 0\end{array}\right]）,\end{array}$$\begin{array}{rl}& w\left(\left[\begin{array}{cccc}{A}_1& A_2& \dots & A_n\\ A_2& \dots & A_n& A_1\\ ⋮& \ddots & \ddots & ⋮\\ A_n& A_1& \dots & A_{n-1}\end{array}\right]\right)\\ & =\underset{1\le k\le n}{max}w\left(\left[\begin{array}{cc}0& {\displaystyle \sum _{m=1}^n{\omega }^{k(m-1)}{A}_m}\\ {\displaystyle \sum _{m=1}^n{\omega }^{(n-k)(m-1)}{A}_m}& 0\end{array}\right]\right),\end{array}$在哪里ω =e2π _ 我n$\omega ={\mathrm{e}}^{\frac{2\pi \text{ 我}}{n}}$$\omega ={\mathrm{e}}^{\frac{2\pi \text{ i}}{n}}$是n次单位根。同时，得到了一般算子矩阵的一些不等式。