Lipschitz

We give conditions that ensure an operator satisfying a Piestch domination in given setting also satisfies different setting. From this we derive bounded multilinear T T is Lipschitz alttext="p"> p encoding="application/x-tex">p -summing if and only the mapping alttext="f Subscript upper T Baseline left-parenthesis x 1 circled-times midline-horizontal-ellipsis n right-parenthesis colon-equal comma ellipsis right-parenthesis"> f ( x 1 ⊗ ⋯<!-- ⋯ <mml:mi>n stretchy="false">) ≔ , …<!-- … </mml:mrow> encoding="application/x-tex">f_T(x_1\otimes \cdots \otimes x_n)≔T(x_1,\ldots , x_n) -summing. The results are based on projective tensor norm. An example with Hilbert norm provided to show statement may not hold when reasonable cross-norm other than considered.