Some basics of a theory of unbounded Wiener–Hopf operators (WH) are developed. The alternative is shown that the domain of a WH is either zero or dense. The symbols for non-trivial WH are determined explicitly by an integrability property. WH are characterized by shift invariance. We study in detail WH with rational symbols showing that they are densely defined, closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains, ranges, spectral and Fredholm points are explicitly determined. Another topic concerns semibounded WH. There is a canonical representation of a semibounded WH using a product of a closable operator and its adjoint. The Friedrichs extension is obtained replacing the operator by its closure. The polar decomposition gives rise to a Hilbert space isomorphism relating a semibounded WH to a singular integral operator of Hilbert transformation type. This remarkable relationship, which allows to transfer results and methods reciprocally, is new also in the thoroughly studied case of bounded WH.

发展了无界Wiener-Hopf算子（WH）的一些理论基础。备选方案显示WH的域为零或密集。非平凡的WH的符号由可积性明确确定。WH的特征在于位移不变性。我们用有理符号对WH进行了详细的研究，表明它们是密集定义的，封闭的并且具有有限维内核和不足空间。明确确定了后者的空间以及域，范围，光谱和Fredholm点。另一个主题涉及半界WH。使用可闭运算符及其伴随乘积的乘积来规范表示半有界的WH。Friedrichs扩展名可以通过关闭操作符来代替。极坐标分解产生了一个希尔伯特空间同构，该半同构将半有界的WH与希尔伯特变换类型的奇异积分算子联系起来。这种无与伦比的关系允许相互转移结果和方法，这在经过全面研究的有界WH情况下也是新的。