Overview
This research focuses on establishing matrix-based versions of comparisons commonly found between $\ell^p$-norms or quasi-norms for sequences of complex numbers. The work extends these comparisons to the context of matrices, particularly involving unitary orbits and spectral order. It provides specific inequalities for sums of normal matrices and applies these findings to concepts such as Olson's spectral order and the relationship between symmetric modulus and quadratic symmetric modulus.
Research Context
The work positions itself within matrix analysis, specifically addressing comparisons analogous to $\ell^p$-norm relationships. A foundational aspect involves extending established properties from sequences of complex numbers to the domain of matrices. This includes exploring the behavior of sums of matrices under specific norms or quasi-norms and how these relate to unitary transformations. The study also builds upon prior majorization results, particularly those attributed to Ando, by providing a completion in the context of positive matrices and their Kato supremum.
Approach
The approach involves establishing specific matrix inequalities. One such inequality is formulated for a family of $m$ normal $d \times d$ matrices, $A_1, \ldots, A_m$. Given $1 \ge q > 0$, the research shows that there exist unitary $d \times d$ matrices $V_1, \ldots, V_d$ such that:
$$ \left|\sum_{k=1}^m A_k\right| \le \frac{1}{d}\sum_{i=1}^d V_i\left\{\sum_{k=1}^m |A_k|^{q}\right\}^{1/q}\,\!\!\!\!V_i^* $$
This inequality provides a matrix version of comparisons involving $\ell^q$-norms. The research then applies these derived matrix inequalities to other mathematical constructs. Specifically, it uses these results to analyze Olson's spectral order and to compare the symmetric modulus with the quadratic symmetric modulus. An additional application involves studying the majorization relationship between the sum of two positive matrices and their Kato supremum.
Findings
- Matrix versions of $\ell^p$-norm and quasi-norm comparisons for sequences of complex numbers have been established.
- For $1 \ge q > 0$ and a family of $m$ normal $d \times d$ matrices $A_1, \ldots, A_m$, it is shown that the inequality $ |\sum_{k=1}^m A_k| \le \frac{1}{d}\sum_{i=1}^d V_i\{\sum_{k=1}^m |A_k|^{q}\}^{1/q}\,\!\!\!\!V_i^* $ holds for some unitary $d \times d$ matrices $V_1, \ldots, V_d$.
- The findings provide applications to Olson's spectral order.
- They also contribute to the comparison between the symmetric modulus and the quadratic symmetric modulus.
- It has been demonstrated that the sum $A+B$ of two positive matrices submajorizes their Kato supremum $A \vee B$.
- This submajorization result completes certain majorization results previously established by Ando.