MULTIPLIERS AND HADAMARD PRODUCTS IN THE VECTOR-VALUED SETTING

Let E-i be Banach spaces, and let X-Ei, be Banach spaces continuously contained in the spaces of E-i-valued sequences ((x) over cap (j))(j) is an element of E-i(N), for i = 1, 2, 3. Given a bounded bilinear map B : E-1 x E-2 -> E-3, we define (X-E2, X-E3)B, the space of B -multipliers between X-E2 and X-E3, to be the set of sequences (lambda(j))(j) is an element of E-1(N) such that (B(lambda(j), (x) over cap (j)))(j) is an element of X-E3 for all ((x) over cap (j))(j) is an element of X-E2, and we define the Hadamard projective tensor product X-E1 circle star(B) X-E2 as consisting of those elements in E-3(N) that can be represented as Sigma(n)Sigma(j) B((x) over cap (n) (j), (y) over cap (n)(j), where (x(n))(n) is an element of X-E1, (y(n))(n) is an element of X-E2, and Sigma(n) parallel to x(n)parallel to X-E1 parallel to y(n)parallel to X-E2 < infinity. We will analyze some properties of these two spaces, relate them, and compute the Hadamard tensor products and the spaces of vector-valued multipliers in several cases, getting applications in the particular case where E = L(E-1, E-2) and B(T, x) = T(x).