正定矩阵与对合：它们的无穷维类似物的情形
Positive definite matrices and involutions: the manners of their infinite cousins
AI总结：
研究\(B(\mathbb{R})\)和\(B(\mathbb{C})\)中矩阵与对合，证明正定对合由正定矩阵共轭相应对合给出，正定矩阵有乔列斯基分解，还构造例子说明其在列向量作用下不一定有特征值及在\(B(\mathbb{C})\)中不一定有平方根。
AI中文摘要：
研究了实（分别地，复）行和列有限的\(\omega\times\omega\)矩阵的代数\(B(\mathbb{R})\)和\(B(\mathbb{C})\)中的矩阵与对合。证明了\(B(\mathbb{R})\)上的任何正定\(\mathbb{R}\)-代数对合（分别地，\(B(\mathbb{C})\)上的任何正定共轭线性对合）由用该代数中的正定矩阵共轭转置对合（分别地，共轭转置对合）给出。这些代数中的所有正定矩阵在代数内都有乔列斯基分解。构造了例子表明这样的正定矩阵对于它们在列有限列向量上的典范作用不一定有任何特征值，并且它们在\(B(\mathbb{C})\)中不一定有任何平方根。
英文摘要：
Matrices and involutions in/on the algebras $B(\mathbb{R})$ and $B(\mathbb{C})$ of real (resp., complex) row- and column-finite $ω\timesω$ matrices are studied. It is proved that any positive definite $\mathbb{R}$-algebra involution on $B(\mathbb{R})$ (resp., any positive definite conjugate-linear involution on $B(\mathbb{C})$) is given by conjugating the transpose involution (resp., the conjugate-transpose involution) with a positive definite matrix from the algebra in question. All positive definite matrices in these algebras have Cholesky factorizations within the algebra. Examples are constructed to show that such positive definite matrices need not have any eigenvalues for their canonical action on column-finite column vectors, and they need not have any square roots in $B(\mathbb{C})$.
