For two given symmetric sequence spaces E and F we study the (E,F)-multiplier space, that is, the space of all matrices M for which the Schur product M ∗ A maps E into F boundedly whenever A does. We obtain several results asserting continuous embedding of the (E,F)-multiplier space into the classical (p,q)-multiplier space (that is, when $E = l_{p}$, $F = l_{q}$). Furthermore, we present many examples of symmetric sequence spaces E and F whose projective and injective tensor products are not isomorphic to any subspace of a Banach space with an unconditional basis, extending classical results of S. Kwapień and A. Pełczyński (1970) and of G. Bennett (1976, 1977) for the case when $E = l_{p}$, $F = l_{q}$.
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Let X, Y be Banach spaces and let 𝓛(X,Y) be the space of bounded linear operators from X to Y. We develop the theory of double operator integrals on 𝓛(X,Y) and apply this theory to obtain commutator estimates of the form $||f(B)S - Sf(A)||_{𝓛(X,Y)} ≤ const||BS - SA||_{𝓛(X,Y)}$ for a large class of functions f, where A ∈ 𝓛(X), B ∈ 𝓛(Y) are scalar type operators and S ∈ 𝓛(X,Y). In particular, we establish this estimate for f(t): = |t| and for diagonalizable operators on $X = ℓ_{p}$ and $Y = ℓ_{q}$ for p < q. We also study the estimate above in the setting of Banach ideals in 𝓛(X,Y). The commutator estimates we derive hold for diagonalizable matrices with a constant independent of the size of the matrix.
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