International Journal of Mathematics and Mathematical Sciences
Volume 29 (2002), Issue 3, Pages 167-178

Compact Hermitian operators on projective tensor products of Banach algebras

T. K. Dutta, H. K. Nath, and H. K. Sarmah

Department of Mathematics, Gauhati University, Guwahati 781014, Assam, India

Received 24 February 2000

Copyright © 2002 T. K. Dutta et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let U and V be, respectively, an infinite- and a finite-dimensional complex Banach algebras, and let UpV 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 UpV 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 UpV is also separable and has HAP; (iii) every compact analytic semigroup (CAS) on U induces the existence of a CAS on UpV having some nice properties. In addition, the converse of the above results are discussed and some open problems are posed.