  In this article, we obtain some results in the direction of ``infinite
dimensional symplectic spectral theory". We prove an inequality between the
eigenvalues and symplectic eigenvalues of a special class of infinite
dimensional operators. Let $T$ be an operator such that $T - \alpha I$ is
compact for some $\alpha > 0$. Denote by $\{{\lambda_j^R}^\downarrow(T)\}$, the
set of eigenvalues of $T$ lying strictly to the right side of $\alpha$ arranged
in the decreasing order and let $\{{\lambda_j^L}^\uparrow(T)\}$ denote the set
of eigenvalues of $T$ lying strictly to the left side of $\alpha$ arranged in
the increasing order. Furthermore, let $\{{d_j^R}^\downarrow(T)\}$ denote the
symplectic eigenvalues of $T$ lying strictly to the right of $\alpha$ arranged
in decreasing order and $\{{d_j^L}^\uparrow(T)\}$ denote the set of symplectic
eigenvalues of $T$ lying strictly to the left of $\alpha$ arranged in
increasing order, respectively (such an arrangement is possible as it will be
shown that the only possible accumulation point for the symplectic eigenvalues
is $\alpha$). Then we show that $${d_j^R}^\downarrow(T) \leq
{\lambda_j^R}^\downarrow(T), \quad j = 1,2, \cdots, s_r$$ and
$${\lambda_j^L}^\uparrow(T) \leq {d_j^L}^\uparrow(T), \quad j = 1,2, \cdots,
s_l,$$ where $s_r$ and $s_l$ denote the number of symplectic eigenvalues of $T$
strictly to the right and left of $\alpha$, respectively. This generalizes a
finite dimensional result obtained by Bhatia and Jain (J. Math. Phys. 56,
112201 (2015)). The class of Gaussian Covariance Operators (GCO) and positive
Absolutely Norm attaining Operators ($(\mathcal{A}\mathcal{N})_+$ operators)
appear as special cases of the set of operators we consider.
