Kernel and range of M(H, K)

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42 3 Means of operators and their comparison

If A is a positive semi-denite matrix, then

                              SA (,)  max aii .                           ()
                                                 i

In fact, this is immediately seen from (); if 1 , . . . , n are the row vectors of A1/2 , then aij (i , j ) for all i, j. (A dierent proof without using () can be found in [4, 42].) Lemma 3.8. An innite matrix A [aij ]i,j1,2, is a Schur multiplier if and only if
sup S[aij ]i,j1, ,n (1,1) . n1

In this case, SA (1,1) is equal to the above supremum.

Proof. If A is a Schur multiplier, then it is obvious that

...

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