标题： 不同希尔伯特空间上作用的算子之间的距离 显示英文标题

标题： Distances between operators acting on different Hilbert spaces

摘要： 本文的目的是定义并比较作用在不同（可分）Hilbert空间上的算子之间的几种距离（或度量）。 我们在这里考虑三种主要情况来测量两个有界算子之间的距离：首先通过取它们的酉轨道之间的距离，其次通过等距嵌入（这推广了Weidmann的一个概念），第三通过准酉等价（使用本文第一作者的概念）。 我们的主要结果是，当算子都是自伴的并且其本质谱中包含$0$时，酉距离和等距距离相等。 此外，对于任何一对有界算子，准酉距离与等距距离是等价的（最多相差一个普遍常数）。 酉距离给出了它们谱的Hausdorff距离的上界。 如果两个算子都有纯本质谱，则酉距离等于它们谱的Hausdorff距离。 通过使用一种尊重离散本征值重数的更精细的谱距离，这种谱距离也等于具有本质谱和离散谱的算子的酉距离。 特别是，上述所有算子距离都等于这个谱距离，或者在自伴算子具有$0$在本质谱中的情况下，由它控制的准酉情形。 我们还通过提出各种（反例）证明了我们的结果是精确的。 最后，我们讨论了相关的收敛概念，补充了我们第一篇论文 arXiv:2202.03234 中的结果。 显示英文摘要

摘要： The aim of this article is to define and compare several distances (or metrics) between operators acting on different (separable) Hilbert spaces. We consider here three main cases of how to measure the distance between two bounded operators: first by taking the distance between their unitary orbits, second by isometric embeddings (this generalises a concept of Weidmann) and third by quasi-unitary equivalence (using a concept of the first author of the present article). Our main result is that the unitary and isometric distances are equal provided the operators are both self-adjoint and have $0$ in their essential spectra. Moreover, the quasi-unitary distance is equivalent (up to a universal constant) with the isometric distance for any pair of bounded operators. The unitary distance gives an upper bound on the Hausdorff distance of their spectrum. If both operators have purely essential spectrum, then the unitary distance equals the Hausdorff distance of their spectra. Using a finer spectral distance respecting multiplicity of discrete eigenvalues, this spectral distance equals the unitary distance also for operators with essential and discrete spectrum. In particular, all operator distances mentioned above are equal to this spectral distance resp. controlled by it in the quasi-unitary case for self-adjoint operators with $0$ in the essential spectrum. We also show that our results are sharp by presenting various (counter-)examples. Finally, we discuss related convergence concepts complementing results from our first article arXiv:2202.03234