Building on MacDonald’s formula for the distance from a rank-one projection to the set of nilpotents in $\mathbb {M}_n(\mathbb {C})$, we prove that the distance from a rank $n-1$ projection to the set of nilpotents in $\mathbb {M}_n(\mathbb {C})$ is $\frac {1}{2}\sec (\frac {\pi }{\frac {n}{n-1}+2} )$. For each $n\geq 2$, we construct examples of pairs $(Q,T)$ where Q is a projection of rank $n-1$ and $T\in \mathbb {M}_n(\mathbb {C})$ is a nilpotent of minimal distance to Q. Furthermore, we prove that any two such pairs are unitarily equivalent. We end by discussing possible extensions of these results in the case of projections of intermediate ranks.

In the present paper, an inverse result of approximation, i.e. a saturation theorem for the sampling Kantorovich operators, is derived in the case of uniform approximation for uniformly continuous and bounded functions on the whole real line. In particular, we prove that the best possible order of approximation that can be achieved by the above sampling series is the order one, otherwise the function being approximated turns out to be a constant. The above result is proved by exploiting a suitable representation formula which relates the sampling Kantorovich series with the well-known generalized sampling operators introduced by Butzer. At the end, some other applications of such representation formulas are presented, together with a discussion concerning the kernels of the above operators for which such an inverse result occurs.

In this paper we discuss the asymptotic distribution of the approximation numbers of the finite sections for a Toeplitz operator $T(a) \in \mathcal{L}(\ell^{p,\mu}_N)$, $1 < p < \infty$ and $\mu \in \mathbb{R}$, with a continuous matrix-valued generating function $a$. We prove that the approximation numbers of the finite sections $T_n(a) = P_n T(a) P_n$ have the $k$-splitting property, provided $T(a)$ is a Fredholm operator on $\ell^{p,\mu}_N$.

The distance from an arbitrary rank-one projection to the set of nilpotent operators, in the space of k × k matrices with the usual operator norm, is shown to be sec(π/(k:+2))/2. This gives improved bounds for the distance between the set of all non-zero projections and the set of nilpotents in the space of k × k matrices. Another result of note is that the shortest distance between the set of non-zero projections and the set of nilpotents in the space of k × k matrices is .