Let $\mathcal{H}$ be an infinite dimensional complex Hilbert space and let $\mathcal{B}\left(\mathcal{H}\right)$ be the algebra of all bounded linear operators on $\mathcal{H}$. For every positive integer k and $A\in \mathcal{B}\left(\mathcal{H}\right)$, the k-numerical range of A is the set $W_k\left(A\right)=\left\{\sum _{j=1}^k⟨A{x}_j,x_j⟩:\{x_1,\dots ,x_k\}\text{is an orthonormal set in }\mathcal{H}\right\}.$ In this note, we show that the closure of $W_k\left(A\right)$ can be written as the convex hull of sets involving the essential numerical range of A and $W_{\ell }\left(A\right)$ for $1\le \ell \le k$. We also show that if $W_k\left(A\right)$ is closed, then $W_{\ell }\left(A\right)$ is also closed for $1\le \ell \le k$.

中文翻译：

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k-数值范围的封闭性

让$\mathcal{H}$是一个无限维复希尔伯特空间，让$\mathcal{乙}\left(\mathcal{H}\right)$是所有有界线性算子的代数$\mathcal{H}$. 对于每个正整数k和$\mathrm{一个}\in \mathcal{乙}\left(\mathcal{H}\right)$, A的k数值范围是集合$W_{ķ}\left(\mathrm{一个}\right)=\left\{\sum _{j=1}^{ķ}⟨\mathrm{一个}{X}_j,X_j⟩：\{X_1,\dots ,X_{ķ}\}\text{是一个正交集在 }\mathcal{H}\right\}.$在本说明中，我们展示了$W_{ķ}\left(\mathrm{一个}\right)$可以写成集合的凸包，包括A的基本数值范围和$W_{\ell }\left(\mathrm{一个}\right)$为了$1\le \ell \le ķ$. 我们还表明，如果$W_{ķ}\left(\mathrm{一个}\right)$是关闭的，那么$W_{\ell }\left(\mathrm{一个}\right)$也关闭$1\le \ell \le ķ$.