Eigenvalue pinching theorems on compact symmetric spaces

We prove two first eigenvalue pinching theorems for Riemannian symmetric spaces (Theorems 1 and 2). As their application, we answer negatively a question raised by Elworthy and Rosenberg, who proposed to show that for every compact simple Lie group $G$ with a bi-invariant Riemannian metric $h$ on $G$ with respect to $-\frac{1}{2} \,B$, $B$ being the Killing form of the Lie algebra $\mathfrak{g}$, the first eigenvalue $\lambda _{1}(h)$ would satisfy

for all orthonormal bases $\{v_{j}\}_{j=1}^{n}$ of tangent spaces of $G$ (cf. Corollary 3). This problem arose in an attempt to give a spectral geometric proof that $\pi _{2}(G)=0$ for a Lie group $G$.