The integration with respect to a vector measure may be applied in order to approximate a function in a Hilbert space by means of a finite orthogonal sequence $\left\{f_i\right\}$ attending to two different error criterions. In particular, if $\Omega \in ℝ$ is a Lebesgue measurable set, $f\in L_2\left(\Omega \right)$, and $\left\{A_i\right\}$ is a finite family of disjoint subsets of $\Omega$, we can obtain a measure ${\mu }_0$ and an approximation $f_0$ satisfying the following conditions: (1) $f_0$ is the projection of the function $f$ in the subspace generated by $\left\{f_i\right\}$ in the Hilbert space $f\in L_2\left(\Omega ,{\mu }_0\right)$. (2) The integral distance between $f$ and $f_0$ on the sets $\left\{A_i\right\}$ is small.

We consider the one-parameter family of linear operators that A. Belleni Morante recently introduced and called $B$-bounded semigroups. We first determine all the properties possessed by a couple $\left(A,B\right)$ of operators if they generate a $B$-bounded semigroup ${\left(Y\left(t\right)\right)}_{t\ge 0}$. Then we determine the simplest further property of the couple $\left(A,B\right)$ which can assure the existence of a $C_0$-semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ such that for all $t\ge 0,f\in D\left(B\right)$ we can write $Y\left(t\right)f=T\left(t\right)Bf$. Furthermore, we compare our result with the previous ones and finally we show how our method allows to improve the theory developed by Banasiak for solving implicit evolution equations.

We study evolution semigroups associated with nonautonomous functional differential equations. In fact, we convert a given functional differential equation to an abstract autonomous evolution equation and then derive a representation theorem for the solutions of the underlying functional differential equation. The representation theorem is then used to study the boundedness and almost periodicity of solutions of a class of nonautonomous functional differential equations.