Compact Hermitian operators on projective tensor products of Banach algebras
Let U and V be, respectively, an infinite- and a finite-dimensional complex Banach algebras, and let U⊗pV be their projective tensor product. We prove that (i) every compact Hermitian operator T1 on U gives rise to a compact Hermitian operator T on U⊗pV having the properties that ‖T1‖=‖T‖ and sp(T1...
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| Format: | Article |
| Language: | English |
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Wiley
2002-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171202004659 |
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| author | T. K. Dutta H. K. Nath H. K. Sarmah |
| author_facet | T. K. Dutta H. K. Nath H. K. Sarmah |
| author_sort | T. K. Dutta |
| collection | DOAJ |
| description | Let U and V be, respectively, an infinite- and a
finite-dimensional complex Banach algebras, and let
U⊗pV be their projective tensor product. We prove
that (i) every compact Hermitian operator T1 on U gives rise to a compact Hermitian operator T on U⊗pV having the properties that ‖T1‖=‖T‖ and sp(T1)=sp(T);
(ii) if U and V are separable and U has
Hermitian approximation property (HAP), then U⊗pV is also separable and has HAP;
(iii) every compact analytic semigroup (CAS) on U induces the existence of a CAS on U⊗pV having some nice properties. In addition, the converse of the above results are discussed and some open problems are posed. |
| format | Article |
| id | doaj-art-6dfeeaa89e1343bb9b1c7223a0154b53 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2002-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-6dfeeaa89e1343bb9b1c7223a0154b532025-02-03T05:43:37ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252002-01-0129316717810.1155/S0161171202004659Compact Hermitian operators on projective tensor products of Banach algebrasT. K. Dutta0H. K. Nath1H. K. Sarmah2Department of Mathematics, Gauhati University, Guwahati 781014, Assam, IndiaDepartment of Mathematics, Gauhati University, Guwahati 781014, Assam, IndiaDepartment of Mathematics, Gauhati University, Guwahati 781014, Assam, IndiaLet U and V be, respectively, an infinite- and a finite-dimensional complex Banach algebras, and let U⊗pV be their projective tensor product. We prove that (i) every compact Hermitian operator T1 on U gives rise to a compact Hermitian operator T on U⊗pV having the properties that ‖T1‖=‖T‖ and sp(T1)=sp(T); (ii) if U and V are separable and U has Hermitian approximation property (HAP), then U⊗pV is also separable and has HAP; (iii) every compact analytic semigroup (CAS) on U induces the existence of a CAS on U⊗pV having some nice properties. In addition, the converse of the above results are discussed and some open problems are posed.http://dx.doi.org/10.1155/S0161171202004659 |
| spellingShingle | T. K. Dutta H. K. Nath H. K. Sarmah Compact Hermitian operators on projective tensor products of Banach algebras International Journal of Mathematics and Mathematical Sciences |
| title | Compact Hermitian operators on projective tensor
products of Banach algebras |
| title_full | Compact Hermitian operators on projective tensor
products of Banach algebras |
| title_fullStr | Compact Hermitian operators on projective tensor
products of Banach algebras |
| title_full_unstemmed | Compact Hermitian operators on projective tensor
products of Banach algebras |
| title_short | Compact Hermitian operators on projective tensor
products of Banach algebras |
| title_sort | compact hermitian operators on projective tensor products of banach algebras |
| url | http://dx.doi.org/10.1155/S0161171202004659 |
| work_keys_str_mv | AT tkdutta compacthermitianoperatorsonprojectivetensorproductsofbanachalgebras AT hknath compacthermitianoperatorsonprojectivetensorproductsofbanachalgebras AT hksarmah compacthermitianoperatorsonprojectivetensorproductsofbanachalgebras |