To every subnormal $m$-variable weighted shift $S$ (with bounded positive weights) corresponds a positive Reinhardt measure $\mu$ supported on a compact Reinhardt subset of $\mathbb C^m$. We show that, for $m \geq 2$, the dimensions of the $1$-st cohomology vector spaces associated with the Koszul complexes of $S$ and its dual ${\tilde S}$ are different if a certain radial function happens to be integrable with respect to $\mu$ (which is indeed the case with many classical examples). In particular, $S$ cannot in that case be similar to ${\tilde S}$. We next prove that, for $m \geq 2$, a Fredholm subnormal $m$-variable weighted shift $S$ cannot be similar to its dual.

Let $A_i$, $B_i$ and $X_i$ $(i=1, 2, \dots, n)$ be operators on a separable Hilbert space. It is shown that if $f$ and $g$ are nonnegative continuous functions on $[0,\infty)$ which satisfy the relation $f(t)g(t) =t$ for all $t$ in $[0,\infty)$, then $$ \Biglvert \,\Bigl|\sum^n_{i=1} A^*_i X_i B_i \Bigr|^r \,\Bigrvert^2 \leq \Biglvert \Bigl( \sum^n_{i=1} A^*_i f (|X^*_i|)^2 A_i \Bigr)^r \Bigrvert \, \Biglvert \Bigl( \sum^n_{i=1} B^*_i g (|X_i|)^2 B_i \Bigr)^r \Bigrvert $$ for every $r>0$ and for every unitarily invariant norm. This result improves some known Cauchy-Schwarz type inequalities. Norm inequalities related to the arithmetic-geometric mean inequality and the classical Heinz inequalities are also obtained.