论文标题
通勤元组的痕迹不平等
A trace inequality for commuting tuple of operators
论文作者
论文摘要
对于通勤$ d $ - 运营商的元组$ \ boldsymbol t $ t $在一个复杂的可分离的希尔伯特空间$ \ mathcal h $上定义，让$ \ big [\！\！ （\ big [t_j^*，t_i \ big] \ big）\！\！\ big）$ [t^*_ j，t_i]的$我们通过在拉普莱斯公式中对标量矩阵的确定性的laplace公式中对对称，定义了$ \ big [\！\！\ big [\！我们证明，$ \ big [\！\！\ big [\ boldsymbol t^*，\ boldsymbol t \ big] \！然后，我们应用Amitsur -Levitzki定理来得出结论，对于任何通勤$ d $ - 元组的$ d $ - 普通运营商，$ \ big [\！\！\！\ big！我们表明，如果$ d $ - tuple $ \ boldsymbol t $是循环的，则是$ \ big [\！\！\！尺寸子空间增加到$ \ MATHCAL H $ - 不会迅速增长，然后是操作员$ \ big [\！\！\！ \ big [\ boldsymbol t^*，\ boldsymbol t \ big] \！\！\ big] $是有限的。此外，给出了此迹线的上限。对于一类通勤$ d $ - 元组，该上限被证明是锋利的。我们猜想了可能在更大的一般性中可能遇到的尖锐束缚，并在许多示例中对其进行了验证。
For a commuting $d$- tuple of operators $\boldsymbol T$ defined on a complex separable Hilbert space $\mathcal H$, let $\big [ \!\!\big [ \boldsymbol T^*, \boldsymbol T \big ]\!\!\big ]$ be the $d\times d$ block operator $\big (\!\!\big (\big [ T_j^* , T_i\big ]\big )\!\!\big )$ of the commutators $[T^*_j , T_i] := T^*_j T_i - T_iT_j^*$. We define the determinant of $\big [ \!\!\big [ \boldsymbol T^*, \boldsymbol T \big ]\!\!\big ]$ by symmetrizing the products in the Laplace formula for the determinant of a scalar matrix. We prove that the determinant of $\big [ \!\!\big [ \boldsymbol T^*, \boldsymbol T \big ]\!\!\big ]$ equals the generalized commutator of the $2d$ - tuple of operators, $(T_1,T_1^*, \ldots, T_d,T_d^*)$ introduced earlier by Helton and Howe. We then apply the Amitsur-Levitzki theorem to conclude that for any commuting $d$ - tuple of $d$ - normal operators, the determinant of $\big [ \!\!\big [ \boldsymbol T^*, \boldsymbol T \big ]\!\!\big ]$ must be $0$. We show that if the $d$- tuple $\boldsymbol T$ is cyclic, the determinant of $\big [ \!\!\big [ \boldsymbol T^*, \boldsymbol T \big ]\!\!\big ]$ is non-negative and the compression of a fixed set of words in $T_j^* $ and $T_i$ -- to a nested sequence of finite dimensional subspaces increasing to $\mathcal H$ -- does not grow very rapidly, then the trace of the determinant of the operator $\big [\!\! \big [ \boldsymbol T^* , \boldsymbol T\big ] \!\!\big ]$ is finite. Moreover, an upper bound for this trace is given. This upper bound is shown to be sharp for a class of commuting $d$ - tuples. We make a conjecture of what might be a sharp bound in much greater generality and verify it in many examples.
