Isometric multipliers ofL
p
(G, X) |
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Authors: | U B Tewari P K Chaurasia |
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Institution: | (1) Department of Mathematics, Indian Institute of Technology, 208 016 Kanpur, India |
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Abstract: | Let G be a locally compact group with a fixed right Haar measure andX a separable Banach space. LetL
p
(G, X) be the space of X-valued measurable functions whose norm-functions are in the usualL
p
. A left multiplier ofL
p
(G, X) is a bounded linear operator onB
p
(G, X) which commutes with all left translations. We use the characterization of isometries ofL
p
(G, X) onto itself to characterize the isometric, invertible, left multipliers ofL
p
(G, X) for 1 ≤p ∞,p ≠ 2, under the assumption thatX is not thel
p
-direct sum of two non-zero subspaces. In fact we prove that ifT is an isometric left multiplier ofL
p
(G, X) onto itself then there existsa y ∃ G and an isometryU ofX onto itself such thatTf(x) = U
(R
y
f)(x). As an application, we determine the isometric left multipliers of L1 ∩L
p
(G, X) and L1 ∩C
0
(G, X) whereG is non-compact andX is not the lp-direct sum of two non-zero subspaces. If G is a locally compact abelian group andH is a separable Hubert space, we define
where г is the dual group of G. We characterize the isometric, invertible, left multipliers ofA
p
(G, H), provided G is non-compact. Finally, we use the characterization of isometries ofC(G, X) for G compact to determine the isometric left multipliers ofC(G, X) providedX
*
is strictly convex. |
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Keywords: | Locally compact group Haar measure Banach space-valued measurable functions isometric multipliers |
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