论文标题
从图到Riesz的连续性
From graph to Riesz continuity
论文作者
论文摘要
我们表明，Hilbert Space中的每个图形连续的无限运算符家族在单方面乘以由适当的单一操作员家族单侧乘法后连续变为Riesz。该结果为弗雷德霍尔姆操作员的图形连续家庭提供了一个简单的定义，我们表明，对于此类家族，该索引与Arxiv的N. Ivanov定义的索引相吻合：2111.15081。该结果还为具有紧凑分辨率的操作员提供了两个推论：（1）具有Riesz拓扑的此类操作员空间与具有图形拓扑的此类操作员的空间之间的身份图是同质的； （2）在希尔伯特束纤维之间作用在束束的适当琐碎的情况下，这类操作员的每个图的连续家庭都在riesz中连续。 对于自我伴侣操作员，应通过共轭代替统一运算符的乘法。通常，与适当的共轭连续riesz连续的自相邻操作员的映射家族无法通过适当的共轭使Riesz连续。我们获得了上述自我接合操作员的“琐碎化”结果的部分类似物，并描述了在一般情况下存在这种琐碎化的障碍。这激发了希尔伯特束极化的概念，我们证明了极化的类似结果。这些结果与伊万诺夫（Ivanov）的最新作品Arxiv：2111.15081密切相关，并为他的某些结果提供了替代证明。然后，我们表明，在对参数空间的较小假设和对既不是正面也不是负面的操作员的次要假设下，总是存在一个琐碎的化，这使得家族riesz的连续。
We show that every graph continuous family of unbounded operators in a Hilbert space becomes Riesz continuous after one-sided multiplication by an appropriate family of unitary operators. This result provides a simple definition of the index for graph continuous families of Fredholm operators, and we show that for such families this index coincides with the index defined by N. Ivanov in arXiv:2111.15081. This result also has two corollaries for operators with compact resolvents: (1) the identity map between the space of such operators with the Riesz topology and the space of such operators with the graph topology is a homotopy equivalence; (2) every graph continuous family of such operators acting between fibers of Hilbert bundles becomes Riesz continuous in appropriate trivializations of the bundles. For self-adjoint operators, multiplication by unitary operators should be replaced by conjugation. In general, a graph continuous family of self-adjoint operators with compact resolvents cannot be made Riesz continuous by an appropriate conjugation. We obtain a partial analogue of the "trivialization" result above for self-adjoint operators and describe obstructions to existence of such a trivialization in the general case. This motivates the notion of a polarization of a Hilbert bundle, and we prove a similar result for polarizations. These results are closely related to the recent work arXiv:2111.15081 of Ivanov and provide alternative proofs for some of his results. We then show that, under a minor assumption on the space of parameters and for operators which are neither essentially positive nor essentially negative, there is always a trivialization making the family Riesz continuous.
