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

TL;DR

This paper establishes an inequality relating eigenvalues and symplectic eigenvalues for a class of infinite-dimensional operators, extending finite-dimensional spectral results and including Gaussian Covariance Operators as special cases.

Contribution

It generalizes finite-dimensional eigenvalue inequalities to infinite-dimensional operators, particularly for operators close to scalar multiples of the identity.

Findings

Proves inequalities between eigenvalues and symplectic eigenvalues for certain operators.

Shows the only accumulation point of symplectic eigenvalues is a specific scalar.

Includes Gaussian Covariance Operators as special cases.

Abstract

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.