An Order Relation between Eigenvalues and Symplectic Eigenvalues of a Class of Infinite Dimensional Operators

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 by considering different cases with respect to the cardinality of the eigenvalues of $T$ 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 ($(\mathcal{AN})_+$ operators) appear as special cases of the set of operators we consider.