  Let $X$, $Y$ be Banach spaces and let $\mathcal{L}(X,Y)$ be the space of
bounded linear operators from $X$ to $Y$. We develop the theory of double
operator integrals on $\mathcal{L}(X,Y)$ and apply this theory to obtain
commutator estimates of the form $\|f(B)S-Sf(A)\|_{\mathcal{L}(X,Y)}\leq
\textrm{const} \|BS-SA\|_{\mathcal{L}(X,Y)}$ for a large class of functions
$f$, where $A\in\mathcal{L}(X)$, $B\in \mathcal{L}(Y)$ are scalar type
operators and $S\in \mathcal{L}(X,Y)$. In particular, we establish this
estimate for $f(t):=|t|$ and for diagonalizable operators on $X=\ell_{p}$ and
$Y=\ell_{q}$, for $p<q$ and $p=q=1$, and for $X=Y=\mathrm{c}_{0}$. We also
obtain results for $p\geq q$. We also study the estimate above in the setting
of Banach ideals in $\mathcal{L}(X,Y)$. The commutator estimates we derive hold
for diagonalizable matrices with a constant independent of the size of the
matrix.

Functional Analysis;Operator Algebras