Integral expression and comparison of norms

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30 2 Double integral transformations

Proposition . Let (s, t) be the Schur multiplier in the previous lemma. If (s, t) is a Schur multiplier relative to a pair (H, K), then so is the point- wise product (s, t) (s, t)(s, t). Furthermore, for each X B(H) the corresponding double integral transformations (X) and (X) are related by

           (X)       (HsH )ix ((X))(KsK )ix d1 (x)

                              (HsH )ix ((X))(1  sK ) d2 (x)

                              (1  sH )((X))(KsK )ix d3 (x)

              ...

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