In this paper we provide a complete description of linear biseparating maps between spaces lip0（X^a, E） of Banach-valued little Lipschitz functions vanishing at infinity on locally com-pact HSlder metric spaces X^a=...In this paper we provide a complete description of linear biseparating maps between spaces lip0（X^a, E） of Banach-valued little Lipschitz functions vanishing at infinity on locally com-pact HSlder metric spaces X^a=（X,dx^a） with 0〈a〈1.Namely, it is proved that any linear bijection T ： lip0（X^a,E）→lip0（Y^a,F）satisfying that ｜｜Tf（y）｜｜F｜｜Tg（y）｜｜F= 0 for all y ∈ Y if and only if ｜｜f（x）｜｜E｜｜g（x）｜｜E=0 for all x E X, is a weighted composition operator of the form Tf（y） = h（y）（f（φ（y）））, where φ is a homeomorphism from Y onto X and h is a map from Y into the set of all linear bijections from E onto F. Moreover, T is continuous if and only if h（y） is continuous for all y ∈ Y. In this case, φ becomes a locally Lipschitz homeomorphism and h a locally Lipschitz map from Y^a into the space of all continuous linear bijections from E onto F with the metric induced by the operator canonical norm. This enables us to study the automatic continuity of T and the existence of discontinuous linear biseparating maps.展开更多
基金supported by Junta de Andalucia grants FQM-1438 and FQM-3737
文摘In this paper we provide a complete description of linear biseparating maps between spaces lip0（X^a, E） of Banach-valued little Lipschitz functions vanishing at infinity on locally com-pact HSlder metric spaces X^a=（X,dx^a） with 0〈a〈1.Namely, it is proved that any linear bijection T ： lip0（X^a,E）→lip0（Y^a,F）satisfying that ｜｜Tf（y）｜｜F｜｜Tg（y）｜｜F= 0 for all y ∈ Y if and only if ｜｜f（x）｜｜E｜｜g（x）｜｜E=0 for all x E X, is a weighted composition operator of the form Tf（y） = h（y）（f（φ（y）））, where φ is a homeomorphism from Y onto X and h is a map from Y into the set of all linear bijections from E onto F. Moreover, T is continuous if and only if h（y） is continuous for all y ∈ Y. In this case, φ becomes a locally Lipschitz homeomorphism and h a locally Lipschitz map from Y^a into the space of all continuous linear bijections from E onto F with the metric induced by the operator canonical norm. This enables us to study the automatic continuity of T and the existence of discontinuous linear biseparating maps.
