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- W4310996968 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 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." @default.
- W4310996968 created "2022-12-22" @default.
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- W4310996968 date "2022-12-07" @default.
- W4310996968 modified "2023-09-30" @default.
- W4310996968 title "An Order Relation between Eigenvalues and Symplectic Eigenvalues of a Class of Infinite Dimensional Operators" @default.
- W4310996968 doi "https://doi.org/10.48550/arxiv.2212.03900" @default.
- W4310996968 hasPublicationYear "2022" @default.
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