{"title":"数算子的逆不等式","authors":"M. Garayev, H. Guediri, N. Altwaijry","doi":"10.30546/2409-4994.48.2.2022.179","DOIUrl":null,"url":null,"abstract":"For a bounded linear operator A on a reproducing kernel Hilbert space H (Ω), with normalized reproducing kernel b k λ = k λ k k λ k , the Berezin symbol, Berezin number and Berezin norm are defined respectively by e A ( λ ) = h A b k λ , b k λ i , ber ( A ) = sup λ ∈ Ω (cid:12)(cid:12)(cid:12) e A ( λ ) (cid:12)(cid:12)(cid:12) and k A k ber = sup λ ∈ Ω (cid:13)(cid:13)(cid:13) A b k λ (cid:13)(cid:13)(cid:13) . A straightforward comparison between these character-istics yields the inequalities ber ( A ) ≤ k A k ber ≤ k A k . In this paper, we prove further inequalities relating them, and give special care to the corresponding reverse inequalities. In particular, we refine the first one of the above inequalities, namely we prove that ber ( A ) ≤ (cid:18) k A k 2 ber − inf λ ∈ Ω (cid:13)(cid:13)(cid:13) ( A − e A ( λ )) b k λ (cid:13)(cid:13)(cid:13) 2 (cid:19) 12 .","PeriodicalId":54068,"journal":{"name":"Proceedings of the Institute of Mathematics and Mechanics","volume":"32 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2021-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"REVERSE INEQUALITIES FOR THE BEREZIN NUMBER OF\\nOPERATORS\",\"authors\":\"M. Garayev, H. Guediri, N. Altwaijry\",\"doi\":\"10.30546/2409-4994.48.2.2022.179\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For a bounded linear operator A on a reproducing kernel Hilbert space H (Ω), with normalized reproducing kernel b k λ = k λ k k λ k , the Berezin symbol, Berezin number and Berezin norm are defined respectively by e A ( λ ) = h A b k λ , b k λ i , ber ( A ) = sup λ ∈ Ω (cid:12)(cid:12)(cid:12) e A ( λ ) (cid:12)(cid:12)(cid:12) and k A k ber = sup λ ∈ Ω (cid:13)(cid:13)(cid:13) A b k λ (cid:13)(cid:13)(cid:13) . A straightforward comparison between these character-istics yields the inequalities ber ( A ) ≤ k A k ber ≤ k A k . In this paper, we prove further inequalities relating them, and give special care to the corresponding reverse inequalities. In particular, we refine the first one of the above inequalities, namely we prove that ber ( A ) ≤ (cid:18) k A k 2 ber − inf λ ∈ Ω (cid:13)(cid:13)(cid:13) ( A − e A ( λ )) b k λ (cid:13)(cid:13)(cid:13) 2 (cid:19) 12 .\",\"PeriodicalId\":54068,\"journal\":{\"name\":\"Proceedings of the Institute of Mathematics and Mechanics\",\"volume\":\"32 1\",\"pages\":\"\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2021-09-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the Institute of Mathematics and Mechanics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.30546/2409-4994.48.2.2022.179\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Institute of Mathematics and Mechanics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30546/2409-4994.48.2.2022.179","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
为有界的线性算子在再生核希尔伯特空间H(Ω),与标准化复制内核b kλ= kλk kλk, Berezin符号,Berezin数量和Berezin规范定义分别通过e(λ)= H b kλ,我kλ,误码率(a) =一口λ∈Ω(cid: 12) (cid: 12) (cid: 12) e(λ)(cid: 12) (cid: 12) (cid: 12)和k = k误码率一口λ∈Ω(cid: 13) (cid: 13) (cid: 13) b kλ(cid: 13) (cid: 13) (cid: 13)。对这些特性的直接比较得出不等式ber (A)≤k A k ber≤k A k。本文进一步证明了与它们有关的不等式，并特别注意了相应的逆不等式。特别地，我们改进了上面的第一个不等式，即证明了ber (A)≤(cid:18) k A k 2 ber−inf λ∈Ω (cid:13)(cid:13)(cid:13) (cid:13))(A−e A (λ)) b k λ (cid:13)(cid:13)(cid:13) 2 (cid:19) 12。
REVERSE INEQUALITIES FOR THE BEREZIN NUMBER OF
OPERATORS
For a bounded linear operator A on a reproducing kernel Hilbert space H (Ω), with normalized reproducing kernel b k λ = k λ k k λ k , the Berezin symbol, Berezin number and Berezin norm are defined respectively by e A ( λ ) = h A b k λ , b k λ i , ber ( A ) = sup λ ∈ Ω (cid:12)(cid:12)(cid:12) e A ( λ ) (cid:12)(cid:12)(cid:12) and k A k ber = sup λ ∈ Ω (cid:13)(cid:13)(cid:13) A b k λ (cid:13)(cid:13)(cid:13) . A straightforward comparison between these character-istics yields the inequalities ber ( A ) ≤ k A k ber ≤ k A k . In this paper, we prove further inequalities relating them, and give special care to the corresponding reverse inequalities. In particular, we refine the first one of the above inequalities, namely we prove that ber ( A ) ≤ (cid:18) k A k 2 ber − inf λ ∈ Ω (cid:13)(cid:13)(cid:13) ( A − e A ( λ )) b k λ (cid:13)(cid:13)(cid:13) 2 (cid:19) 12 .
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Proceedings of the Institute of Mathematics and Mechanics (PIMM), National Academy of Sciences of Azerbaijan is an open access journal that publishes original, high quality research papers in all fields of mathematics. A special attention is paid to the following fields: real and complex analysis, harmonic analysis, functional analysis, approximation theory, differential equations, calculus of variations and optimal control, differential geometry, algebra, number theory, probability theory and mathematical statistics, mathematical physics. PIMM welcomes papers that establish interesting and important new results or solve significant problems. All papers are refereed for correctness and suitability for publication. The journal is published in both print and online versions.
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