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These are problems from Example 1.4.2 of Fukushima's book "Dirichlet forms and symmetric Markov processes".

Let $\Lambda$ be the set of all real sequences $\left\{\lambda_n\right\}_{n\in\mathbf{Z}}$ satisfying $$\lambda_0=0, \lambda_n=\lambda_{-n}\text{ for all }n\in\mathbf{Z},$$ and $$\sum_{m,n\in\mathbf{Z}}(\lambda_n+\lambda_m-\lambda_{n-m})\rho_n\rho_m\ge0,$$ where $\left\{\rho_n\right\}_{n\in\mathbf{Z}}$ is any sequence of real numbers with only finite elements non-vanishing.

  1. The authors claim that if $\left\{\lambda_n\right\}$ satisfies the first condition and $\lambda_n$ is concave and increasing for $n>0$, then it satisfies the second condition.

  2. The authors claim that if $\left\{\lambda_n\right\}\in\Lambda$, then for all $t>0$ $$\sum_{m,n\in\mathbf{Z}}\exp[t(\lambda_n+\lambda_m-\lambda_{n-m})]\rho_n\rho_m\ge0,$$ where $\left\{\rho_n\right\}_{n\in\mathbf{Z}}$ is any sequence of real numbers with only finite elements non-vanishing.

I do not know how to prove these two results. Could someone give some hints?

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  • $\begingroup$ 2) follows from 1). Indeed, pointwise product of positive semidefinite matrices $A,B$ is again p.d., hence so are $A^k$ for $k\geq 0$ and $e^{tA}$ for $t>0$. $\endgroup$
    – Fedor Petrov
    Commented Feb 2, 2016 at 0:10
  • $\begingroup$ For proving 1), it suffices to consider $\lambda_n=\min(|n|,M)$ for fixed positive integer $M$. Any other concave increasing sequence is a positive linear combination of such sequences. $\endgroup$
    – Fedor Petrov
    Commented Feb 2, 2016 at 0:12
  • $\begingroup$ But how to prove this case? $\endgroup$
    – yangmengqh
    Commented Feb 2, 2016 at 7:35

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