论文标题
Rochberg的抽象串联定理重新审视了
Rochberg's abstract coboundary theorem revisited
论文作者
论文摘要
Rochberg的Cooboundarary定理提供了方程$（I-T）y = X $的条件，可在$ y $中解决。在这里，$ t $是希尔伯特空间的单方面转变，$ i $是身份操作员，$ x $是给定的矢量。条件是根据由$ t $确定的Wold型分解和$ x $ $ t $的增长来表示的。我们重新访问罗奇伯格的定理，并证明以下结果。令$ t $为在Hilbert Space $ \ Mathcal {H} $上作用的等轴测图，让$ x \ in \ Mathcal {H} $。假设$ \ sum_ {k = 0}^\ infty k \ | t^{*k} x \ | <\ infty $。然后，$ x $在$（i-t）$的范围内，如果（仅当）$ \ | \ sum_ {k = 0}^n t^k x \ | = o（\ sqrt {n}）。$当$ t $仅仅是收缩时，$ x $是在附加假设下的串联。给出了Fortet和KAC考虑的功能方程式$ f（x）-f（2x）-f（2x）-f（2x）-f（2x）-f（x）-f（2x）= F（x）$的某些应用。
Rochberg's coboundary theorem provides conditions under which the equation $(I-T)y = x$ is solvable in $y$. Here $T$ is a unilateral shift on Hilbert space, $I$ is the identity operator and $x$ is a given vector. The conditions are expressed in terms of Wold-type decomposition determined by $T$ and growth of iterates of $T$ at $x$. We revisit Rochberg's theorem and prove the following result. Let $T$ be an isometry acting on a Hilbert space $\mathcal{H}$ and let $x \in \mathcal{H}$. Suppose that $\sum_{k=0}^\infty k \| T^{*k} x \| < \infty$. Then $x$ is in the range of $(I-T)$ if (and only if) $\|\sum_{k= 0}^n T^k x \| = o(\sqrt{n}).$ When $T$ is merely a contraction, $x$ is a coboundary under an additional assumption. Some applications to $L^2$-solutions of the functional equation $f(x)-f(2x) = F(x)$, considered by Fortet and Kac, are given.
