Beberapa Sifat Operator Hilbert-SchmidtPada Ruang L2 ( M )
Muslim Ansori · 2007
Let M 1 ⊂ Rm and M 2 ⊂ Rn be measurable sets, respectively. A continuous linear operator K : L 2 ( M1 ) → L2 ( M 2 ) is a Hilbert-Schmidt operator if and only if there is a function k ∈ L 2 ( M 2 × M 1 ) such that ( Kf )( x ) = ∫ M1 k ( x, y ) f ( y ) dy almost everywhere for every f ∈ L 2 ( M 2 ) ; the function k is called kernel of K . If K is the norm of K , then K = ( ∫ M 2 ∫ M1 k ( x , y ) 2 dydx ) 1 2 = k The collection of the Hilbert-Schmidt operator from L 2 ( M 1 ) into L 2 ( M 2 ) is denoted by Further, we shal show that K ( M , M ) , M ⊂ Rn , is ∗ -algebra in L c ( M , M ) . Keywords : involution, adjoint, ∗ -algebra