论文标题
$ \ boldsymbol {s} _p，〜0 <p <1，$和相关不平等的三角投影
Triangular projection on $\boldsymbol{S}_p,~0<p<1,$ and related inequalities
论文作者
论文摘要
在本文中，我们研究了$ n \ times n $矩阵的三角投影$ {\ Mathcal p} _n $的属性。投影$ {\ Mathcal P} _n $歼灭了主角下方的$ n \ times n $矩阵的条目，并使其余条目保持不变。我们估计$ {\ mathcal p} _n $的$ p $ - norms作为schatten-von neumann类$ \ boldsymbol {s} _p $ for $ 0 <p <1 $的操作员。本文的主要结果表明，对于$ p \ in（0,1）$，$ {\ mathcal p} _n $的$ p $ - norms上的$ \ boldsymbol {s} _p $以$ n \ to \ infty $为$ n^{1/p-1} $。这解决了B.S.提出的问题喀什。 在本文的其他结果中，我们提到的结果描述了$ \ boldsymbol {s} _p $ - Quasinorms的行为，其$ n \ times n $矩阵的矩阵上方的条目等于1，而对角线以下的条目则等于0。
In this paper we study properties of the triangular projection ${\mathcal P}_n$ on the space of $n\times n$ matrices. The projection ${\mathcal P}_n$ annihilates the entries of an $n\times n$ matrix below the main diagonal and leaves the remaining entries unchanged. We estimate the $p$-norms of ${\mathcal P}_n$ as an operator on the Schatten--von Neumann class $\boldsymbol{S}_p$ for $0<p<1$. The main result of the paper shows that for $p\in(0,1)$, the $p$-norms of ${\mathcal P}_n$ on $\boldsymbol{S}_p$ behave as $n\to\infty$ as $n^{1/p-1}$. This solves a problem posed by B.S. Kashin. Among other results of this paper we mention the result that describes the behaviour of the $\boldsymbol{S}_p$-quasinorms of the $n\times n$ matrices whose entries above the diagonal are equal to 1 while the entries below the diagonal are equal to 0.
