Trace-class norm multipliers

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In this paper we study the pointwise multipliers of the space L 2 ( X , ν ) ⊗ ^ L 2 ( Y , μ ) {L^2}(X,\nu )\hat \otimes {L^2}(Y,\mu ) for different measure spaces ( X , ν ) (X,\nu ) and ( Y , μ ) (Y,\mu ) . First we consider the case where X and Y are finite sets, then we use this to study the multipliers of L 2 ( I , ν ) ⊗ ^ L 2 ( I , ν ) {L^2}(I,\nu )\hat \otimes {L^2}(I,\nu ) , where I is the unit interval and ν \nu any Borel measure on I.