论文标题
有限运算符空间中的一个近似问题
An approximation problem in the space of bounded operators
论文作者
论文摘要
对于Banach空间$ x，y，$，我们考虑有界线性操作员$ \ MATHCAL {l}（x，y）的距离问题。 hand，将全局近似与局部近似有关： \ [d（t，\ mathcal {l}（x，z））= \ sup \ {d（tx，z）：x \ in x，\ | x \ | = 1 \}。 在某些情况下，我们表明，至上是在相应的单位球的极端达到的。此外，当以下等价成立时，我们会得到一些情况： $ t t \ perp_b \ Mathcal {l}（x，z）\ leftrightarrow t^{**} x_0^{**} \ perp_b z^{\ perp \ perp \ perp} \ leftrightArrow t^{**} \ perp_b \ mathcal {l}（x^{**}，z^{\ perp \ perp}），$ x_0^{**} \ in x^{**} $满意$ \ |有限订单的多平滑操作员。另一种情况是$ x $是抽象的$ l_1- $空间，而$ t $是有限订单的多平滑运算符。最后，由于结果，我们获得了足够的条件，即$ y中的子空间$ z $。$。
For Banach spaces $X,Y,$ we consider a distance problem in the space of bounded linear operators $\mathcal{L}(X,Y).$ Motivated by a recent paper \cite{RAO21}, we obtain sufficient conditions so that for a compact operator $T\in\mathcal{L}(X,Y)$ and a closed subspace $Z\subset Y,$ the following equation holds, which relates global approximation with local approximation: \[d(T,\mathcal{L}(X,Z))=\sup\{d(Tx,Z):x\in X,\|x\|=1\}.\] In some cases, we show that the supremum is attained at an extreme point of the corresponding unit ball. Furthermore, we obtain some situations when the following equivalence holds: $$T\perp_B \mathcal{L}(X,Z)\Leftrightarrow T^{**}x_0^{**}\perp_B Z^{\perp\perp}\Leftrightarrow T^{**}\perp_B\mathcal{L}(X^{**},Z^{\perp\perp}),$$ for some $x_0^{**}\in X^{**}$ satisfying $\|T^{**}x_0^{**}\|=\|T^{**}\|\|x_0^{**}\|,$ where $Z^\perp$ is the annihilator of $Z.$ One such situation is when $Z$ is an $L^1-$predual space and an $M-$ideal in $Y$ and $T$ is a multi-smooth operator of finite order. Another such situation is when $X$ is an abstract $L_1-$space and $T$ is a multi-smooth operator of finite order. Finally, as a consequence of the results, we obtain a sufficient condition for proximinality of a subspace $Z$ in $Y.$
