Abstract
LetH be a separable Hilbert space. Every bounded,n-linear operatorL onH n toH(n=0,1,2,…) is shown to have a unique matrix representation with respect to each complete orthonormal sequence {ϕk} 1 ∞ . Conversely, every operator onH n toH possessing a matrix representation is proved to be a bounded,n-linear operator. The foregoing conclusions then apply to polynomial operatorsP onH toH wherePx=L 0+L1x+L2x2+…+Lnxn and eachL k is ak-linear operator.
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