Let $T_k$ be a sequence of $n\times n$ positive matrices such that

Question

Let $\{T_k\}$ be a sequence of $n\times n$ positive matrices such that $$\lim\limits_{N\to\infty}\frac{1}{N}\sum\limits_{k=1}^N T_k=0.$$

I want to see whether this ensures existence of a sub-sequence $T_{n_k}$ which converges to $0$. By this post, I know that this result holds for non-negative matrices real numbers. This result does not hold for arbitrary $C^*$-algebra. I want to see what happens for positive matrices. By positive matrix $T$, I mean that $\langle Th,h\rangle\ge 0$ for all $h$.

What I have tried is as follows: Let us denote $S_N=\frac{1}{N}\sum\limits_{k=1}^N T_k$ and let $\{e_1,\ldots,e_n\}$ be the standard orthonormal basis of $\mathbb{C}^n$. Then

$$\langle S_Ne_1,e_1\rangle =\frac{1}{N}\sum\limits_{k=1}^N \langle T_k e_1,e_1\rangle$$

Then there exists a subsequence $T_{n_k}$ such that $\langle T_{n_k} e_1,e_1\rangle\to0$. Now, I want to get a subsequence of this $T_{n_k}$ (say $T_{m_l}$) such that $\langle T_{m_l} e_2,e_2\rangle\to0$. This way I want to get a final subsequence (say $T_{i_j}$) of $T_k$ such that $\langle T_{i_j} e_k,e_k\rangle\to0$ for all $k$ by induction. But the problem is the Cesaro sum of $T_{n_k}$ may not converge to $0$. This is creating the main issue while applying the induction.

Can anyone help me with any kind of idea or hint? Thanks for your help in advance.

Answer 1

Think about $\langle Te_1,e_1\rangle+\langle Te_2,e_2\rangle+\ldots+\langle Te_n,e_n\rangle$. Remember, it is a fact that for any positive-definite matrix, the entry with the largest absolute value must be on the diagonal and furthermore all the diagonal entries are positive.

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Since $\dfrac{\text{trace}(T_1)+\text{trace}(T_2)+\ldots+\text{trace}(T_N)}{N}\to_{N\to\infty}0$, you get some subsequence $T_{j_k}$ such that $\text{trace}(T_{j_k})\to_{k\to\infty}0$.