spectrum point of are the same (see Proposition \[spectrum-autonomous\]).

Note also that , as well as its time independent version , is an eigenvalue problem in regular sense for $i=1, 2, 3$, as we can see from Definition \[PSP-def\] (1). Many properties of the principal spectrum point/principal eigenvalue have been studied extensively (see [@BaZh; @Cov; @HuShVi; @RaSh; @ShXi1; @ShZh0] etc.). And we will recall the basic properties and prove some new properties of the principal spectrum point/principal eigenvalue of in subsection 3.3.

However, little is known about the eigenvalue problem . We are the first to study the principal spectrum point/principal eigenvalue of nonlocal dispersal operators with time periodic indefinite weight function. Definition \[PSP-def\] (2) is a natural generalization of the principal eigenvalue of random dispersal operators with time periodic indefinite weight functions. And the necessary and sufficient conditions for the existence of positive principal spectrum point of are in Theorem \[PSP-diri\]-\[PSP-peri\], and the properties of the principal spectrum point of are in Theorem \[upper-bounds\].

For $i=1, 2, 3$, let \[hat-mi\] m\_i(x)=\_0\^T m\_i(t,x)dt, \[mi-max-min\] m\_[i, ]{}=\_[x|D\_i]{}m\_i(x), m\_[i, ]{}=\_[x|D\_i]{}m\_i(x), and \[P\] P(m\_i)=\_0\^T\_[x|D\_i]{}m\_i(t, x)dt.

We now state the main results of the paper. We first state the main results on the necessary 