tailieunhanh - Báo cáo toán học: "Schur multiplication on ${\cal B}(\ell_p, \ell_q)$ "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Schur nhân trên $ {\ cal B} (\ ell_p, \ ell_q) $. | J. OPERATOR THEORY 5 1981 231-243 Copyright by INCREST 1981 SCHUR MULTIPLICATION ON QUENTIN F. STOUT 1. INTRODUCTION In 1911 Schur 17 proved that if íỉ j and bij are bounded matrix operators on if 2 then so is aijbij and Ị cfijiy ij ll We call this termwise product Schur multiplication although it is more often called Hadamard multiplication . This term apparently originated in Halmos 4 as a parallel to the Hada-mard product of series. Recently Bennett 2 extended Schur s result to show that for 1 p Ợ oo Schur multiplication gives a commutative Banach algebra structure to the bounded matrix operators from fp to This paper studies these Banach algebras exhibiting their maximal ideal spaces and some of their properties. If p q the Schur-Bennett results are startling for they have taken a highly noncommutative Banach algebra and given it a nontrivial commutative multiplication consistant with the original norm and linear structure. Varopoulos 24 is interested in such compatible Banach algebras and has asked if Schur multiplication on is a 2-algebra. If p q Schur multiplication accomplishes even more by supplying a multiplication to a collection of operators on which there is no natural product. For these reasons alone it is an interesting subject of study but there is more. Schur multiplication is useful in many areas of linear algebra analysis and statistics and there is an increasing awareness of its role. It has been used in operator theory Halmos and Sunder 5 Johnson and Williams 7 Shields and Wallen 20 complex analysis FitzGerald and Horn 3 Pommerenke 14 Shapiro and Shields 18 19 Banach spaces Bennett 1 Kwapien and Pelczyiiski 9 and combinatorics Ryser 16 . Further Styan 23 has a survey article outlining its uses in multivariate analysis Bennett 2 uses it to unify and improve results on absolutely summing operators and Stout 21 22 explores connections with the essential numerical range and interpolating ideals. We begin in Section 2 by introducing some notation .