An Order Relation between Eigenvalues and Symplectic Eigenvalues of a Class of Infinite-Dimensional Operators
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| Formato: | Preprint |
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2022
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| author | John, Tiju Cherian Kumar, V. B. Kiran Tonny, Anmary |
| author_facet | John, Tiju Cherian Kumar, V. B. Kiran Tonny, Anmary |
| contents | 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 $T - αI$ is compact for some $α> 0$. Denote by $\{{λ_j^R}^\downarrow(T)\}$, the set of eigenvalues of $T$ lying strictly to the right side of $α$ arranged in the decreasing order and let $\{{λ_j^L}^\uparrow(T)\}$ denote the set of eigenvalues of $T$ lying strictly to the left side of $α$ arranged in the increasing order. Furthermore, let $\{{d_j^R}^\downarrow(T)\}$ denote the symplectic eigenvalues of $T$ lying strictly to the right of $α$ arranged in decreasing order and $\{{d_j^L}^\uparrow(T)\}$ denote the set of symplectic eigenvalues of $T$ lying strictly to the left of $α$ 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 $α$). Then we show that $${d_j^R}^\downarrow(T) \leq {λ_j^R}^\downarrow(T), \quad j = 1,2, \cdots, s_r$$ and $${λ_j^L}^\uparrow(T) \leq {d_j^L}^\uparrow(T), \quad j = 1,2, \cdots, s_l,$$ where $s_r$ and $s_l$ denote the number of symplectic eigenvalues of $T$ strictly to the right and left of $α$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_03900 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | An Order Relation between Eigenvalues and Symplectic Eigenvalues of a Class of Infinite-Dimensional Operators John, Tiju Cherian Kumar, V. B. Kiran Tonny, Anmary Spectral Theory Quantum Physics 47A10, 47B07, 81S10 (Primary), 46L60, 47B15, 47B65 (Secondary) 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 $T - αI$ is compact for some $α> 0$. Denote by $\{{λ_j^R}^\downarrow(T)\}$, the set of eigenvalues of $T$ lying strictly to the right side of $α$ arranged in the decreasing order and let $\{{λ_j^L}^\uparrow(T)\}$ denote the set of eigenvalues of $T$ lying strictly to the left side of $α$ arranged in the increasing order. Furthermore, let $\{{d_j^R}^\downarrow(T)\}$ denote the symplectic eigenvalues of $T$ lying strictly to the right of $α$ arranged in decreasing order and $\{{d_j^L}^\uparrow(T)\}$ denote the set of symplectic eigenvalues of $T$ lying strictly to the left of $α$ 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 $α$). Then we show that $${d_j^R}^\downarrow(T) \leq {λ_j^R}^\downarrow(T), \quad j = 1,2, \cdots, s_r$$ and $${λ_j^L}^\uparrow(T) \leq {d_j^L}^\uparrow(T), \quad j = 1,2, \cdots, s_l,$$ where $s_r$ and $s_l$ denote the number of symplectic eigenvalues of $T$ strictly to the right and left of $α$, 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. |
| title | An Order Relation between Eigenvalues and Symplectic Eigenvalues of a Class of Infinite-Dimensional Operators |
| topic | Spectral Theory Quantum Physics 47A10, 47B07, 81S10 (Primary), 46L60, 47B15, 47B65 (Secondary) |
| url | https://arxiv.org/abs/2212.03900 |