论文标题
绝对最小达到无界正常运算符的光谱表示
Spectral representation of absolutely minimum attaining unbounded normal operators
论文作者
论文摘要
令$ t：d（t）\ rightarrow h_2 $是一个密集定义的闭合操作员，域$ d（t）\ subset h_1 $。我们说，如果对于每个闭合子空间$ m $ $ h_1 $，限制操作员$ t | _m：d（t）\ cap m \ rightArrow h_2 $都达到其最小模量$ m（t | _ {m}）$，则说明$ t $绝对达到最低。也就是说，在d（t）\ cap m $中存在$ x \ with $ \ | x \ | = 1 $和$ \ | t（x）\ | = \ inf \ {\ | t（m）\ |：m \ in d（t）\ cap m：\ | m \ | = 1 \} $。在本文中，我们证明了这类运营商的几个特征，并表明该类别的每个运营商都有一个非平凡的超级传播子空间。我们还证明了该类别无界正常运算符的光谱定理。事实证明，每个此类操作员都有紧凑的分解。
Let $T:D(T)\rightarrow H_2$ be a densely defined closed operator with domain $D(T)\subset H_1$. We say $T$ to be absolutely minimum attaining if for every closed subspace $M$ of $H_1$, the restriction operator $T|_M:D(T)\cap M\rightarrow H_2$ attains its minimum modulus $m(T|_{M})$. That is, there exists $x \in D(T)\cap M$ with $\|x\|= 1$ and $\|T(x)\| = \inf \{\|T(m)\|: m \in D(T) \cap M: \|m\|=1\}$. In this article, we prove several characterizations of this class of operators and show that every operator in this class has a nontrivial hyperinvariant subspace. We also prove a spectral theorem for unbounded normal operators of this class. It turns out that every such operator has a compact resolvent.
