On the boundedness of quaternionic numerical ranges with respect to nonstandard involutions
Let ϕ be a nonstandard involution on the set of all quaternion numbers, and α be a quaternion such that ϕ(α)=α. For any n×n quaternion matrix A, the notion of numerical range of A with respect to ϕ, Wϕ(α)(A), was introduced by L. Rodman in 2014, and for the case α=0, he characterized all quaternion...
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| Veröffentlicht in: | Linear algebra and its applications 2021-02, Vol.610, p.59-72 |
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| creator | Rahjoo, Meysam Aghamollaei, Gholamreza Momenaee Kermani, Hossein |
| description | Let ϕ be a nonstandard involution on the set of all quaternion numbers, and α be a quaternion such that ϕ(α)=α. For any n×n quaternion matrix A, the notion of numerical range of A with respect to ϕ, Wϕ(α)(A), was introduced by L. Rodman in 2014, and for the case α=0, he characterized all quaternion matrices A whose Wϕ(0)(A) is bounded. In this paper, for the case α≠0, those quaternion matrices A for which Wϕ(α)(A) is bounded, are characterized. Also, by introducing the notion of ϕ-class of a quaternion number, all quaternion matrices A whose Wϕ(α)(A) is a singleton set, are characterized, and for n≥3, it is shown that Wϕ(α)(A) is bounded if and only if it is a singleton set. |
| doi_str_mv | 10.1016/j.laa.2020.09.036 |
| format | Article |
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Also, by introducing the notion of ϕ-class of a quaternion number, all quaternion matrices A whose Wϕ(α)(A) is a singleton set, are characterized, and for n≥3, it is shown that Wϕ(α)(A) is bounded if and only if it is a singleton set.</description><subject>Linear algebra</subject><subject>Nonstandard involution</subject><subject>Numerical range</subject><subject>Quaternion matrices</subject><subject>Quaternions</subject><issn>0024-3795</issn><issn>1873-1856</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9kEtLAzEQgIMoWB8_wFvAc9dJsq_gSYovKPSiJw8hm8zalJq0SVbx3xupZy8zDMw3j4-QKwYVA9bebKqt1hUHDhXICkR7RGas78Sc9U17TGYAvJ6LTjan5CylDQDUHfAZeVt5mtdIhzB5i9ZjSjSMdD_pjNG74J2hfvrA6Ize0qj9Oyb65fKaRkw7NJnmQH3wKWtvdbTU-c-wnXIh0wU5GfU24eVfPievD_cvi6f5cvX4vLhbzo3gTS6x7w2yegRrRDPUULcNt7KUupFy6Pp-1J1BLvvBINaMSWl5ZzscBy4G3Ylzcn2Yu4thP2HKahOm6MtKxRsGLYimFqWLHbpMDClFHNUuug8dvxUD9etQbVRxqH4dKpCqOCzM7YHBcv6nw6iScegNWhfL78oG9w_9A6Pke3A</recordid><startdate>20210201</startdate><enddate>20210201</enddate><creator>Rahjoo, Meysam</creator><creator>Aghamollaei, Gholamreza</creator><creator>Momenaee Kermani, Hossein</creator><general>Elsevier Inc</general><general>American Elsevier Company, Inc</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><orcidid>https://orcid.org/0000-0001-7731-4137</orcidid></search><sort><creationdate>20210201</creationdate><title>On the boundedness of quaternionic numerical ranges with respect to nonstandard involutions</title><author>Rahjoo, Meysam ; Aghamollaei, Gholamreza ; Momenaee Kermani, Hossein</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c325t-c388ce14f0dc35b404652d9f0da599b788fa7ce298bcee41199d27d7efb23ba73</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Linear algebra</topic><topic>Nonstandard involution</topic><topic>Numerical range</topic><topic>Quaternion matrices</topic><topic>Quaternions</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Rahjoo, Meysam</creatorcontrib><creatorcontrib>Aghamollaei, Gholamreza</creatorcontrib><creatorcontrib>Momenaee Kermani, Hossein</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Linear algebra and its applications</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Rahjoo, Meysam</au><au>Aghamollaei, Gholamreza</au><au>Momenaee Kermani, Hossein</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the boundedness of quaternionic numerical ranges with respect to nonstandard involutions</atitle><jtitle>Linear algebra and its applications</jtitle><date>2021-02-01</date><risdate>2021</risdate><volume>610</volume><spage>59</spage><epage>72</epage><pages>59-72</pages><issn>0024-3795</issn><eissn>1873-1856</eissn><abstract>Let ϕ be a nonstandard involution on the set of all quaternion numbers, and α be a quaternion such that ϕ(α)=α. 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| subjects | Linear algebra Nonstandard involution Numerical range Quaternion matrices Quaternions |
| title | On the boundedness of quaternionic numerical ranges with respect to nonstandard involutions |
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