Isometries of certain operator spaces

Abstract

Abstract. Let X and Y be Banach spaces, and L(X, Y ) be the spaces of bounded linear operators from X into Y. In this paper we give full characterization of isometric onto operators of L(X, Y ), for a certain class of Banach spaces, that includes p , 1 < p < ∞. We also characterize the isometric onto operators of L(c 0 ) and K( 1 ), the compact operators on 1 . Furthermore, the multiplicative isometric onto operators of L( 1 ), when multiplication on L( 1 ) is taken to be the Schur product, are characterized. IntroductionLet X and Y be Banach spaces, and L(X, Y ) the space of bounded linear oper- [12], it was proved that every isometry of a semisimple commutative Banach algebra that preserves the identity is multiplicative. In this paper, we give a full characterization of isometric onto operators of L(X, Y ), for a class of Banach spaces that includes p , 1 < p < ∞. Isometric onto operators of L(c 0 ), and K( 1 ), the compact operators on 1 , are also fully characterized in this paper. Furthermore, multiplicative isometric onto operators of L( 1 ) when multiplication on L( 1 ) is taken to be the Schur product are characterized.Throughout this paper, if X is a Banach space, X * is the dual ofwhere the sum is a direct summand, in the sense, if

Nice surjections on spaces of operators

“…We show that for any Banach space X, a nice operator on L (X, ℓ ∞ ) maps K (X, c 0 ) to itself. This extends the correct part of Theorem 2.1 of [8].…”

“…We now give a partial answer to the necessary condition for nice surjections. The formulation and its proof are based on the proof of Theorem 1.1 in [8]. This result implies automatic weak * -continuity of extreme point-preserving maps on certain domains.…”

“…We also formulate the corresponding result for L (X,Y ) thereby proving an analogue of the result from [9] for L p (1 < p = 2 < ∞) spaces. We also formulate results when nice operators are not of the canonical form, extending and correcting the results from [8].…”

Projections on tensor products of Banach spaces

“…Theorems by Khalil and Saleh [8,9] state that surjective isometries on a class of projective tensor products are dyadic. A theorem by Jarosz states that surjective isometries, that are not reflections, on a class of injective tensor products are also dyadic.…”

“…Khalil in [9], Khalil-Salem in [8], and Jarosz in [11] provided classifications of surjective isometries for different tensor products of Banach spaces that assure the existence of spaces with such isometries. The structure of norm hermitian operators allows an easy characterization of those operators that are also hermitian projections.…”