In this paper we consider various aspects of generalized invertibility of the operator matrixacting on a Banach space$X$⊕$Y$.

Let $ \gamma = \frac{1}{2}\left( {1 + \sqrt {5} } \right) $ denote the golden ratio. H. Davenport and W. M. Schmidt showed in 1969 that, for each non-quadratic irrational real number ξ, there exists a constant$c$> 0 with the property that, for arbitrarily large values of$X$, the inequalities $ \left| {{x_0}} \right| \leqslant X,\,\,\,\left| {{x_0}\xi - {x_1}} \right| \leqslant c{X^{{{{ - 1}} \left/ {\gamma } \right.}}}\,\,\,{\text{and}}\,\,\,\left| {{x_0}{\xi^2} - {x_2}} \right| \leqslant c{X^{{{{ - 1}} \left/ {\gamma } \right.}}} $ admit no non-zero solution $ \left( {{x_0},{x_1},{x_2}} \right) \in {\mathbb{Z}^3} $ . Their result is best possible in the sense that, conversely, there are countably many non-quadratic irrational real numbers ξ such that, for a larger value of c, the same inequalities admit a non-zero integer solution for each$X$≥ 1. Such$extremal$numbers are transcendental and their set is stable under the action of $ {\text{G}}{{\text{L}}_2}\left( \mathbb{Z} \right) $ on $ \mathbb{R}\backslash \mathbb{Q} $ by linear fractional transformations. In this paper, it is shown that there exist extremal numbers ξ for which the Lagrange constant ν(ξ) = liminf_{$q$→∞}$q$||$q$ξ|| is $ \frac{1}{3} $ , the largest possible value for a non-quadratic number, and that there is a natural bijection between the $ {\text{G}}{{\text{L}}_2}\left( \mathbb{Z} \right) $ -equivalence classes of such numbers and the non-trivial solutions of Markoff’s equation.

In this paper, we prove the DonohoStark uncertainty principle for locally compact quantum groups and characterize the minimizer which are bi-shifts of group-like projections. We also prove the HirschmanBeckner uncertainty principle for compact quantum groups and discrete quantum groups. Furthermore, we show Hardy's uncertainty principle for locally compact quantum groups in terms of bi-shifts of group-like projections.

In this article, we classify all standard invariants that can arise from a composed inclusion of an A3with an A4subfactor. More precisely, if N⊂P is an A3subfactor and P⊂M is an A4subfactor, then only four standard invariants can arise from the composed inclusion N⊂M. We answer a question posed by Bisch and Haagerup in 1994. The techniques of this paper also show that there are exactly four standard invariants for the composed inclusion of two A4subfactors.

For the classical space of functions with bounded mean oscillation, it is well known that $\operatorname{VMO}^{**} = \operatorname{BMO}$ and there are many characterizations of the distance from a function$f$in $\operatorname{BMO}$ to $\operatorname{VMO}$ . When considering the Bloch space, results in the same vein are available with respect to the little Bloch space. In this paper such duality results and distance formulas are obtained by pure functional analysis. Applications include general Möbius invariant spaces such as$Q$_{$K$}-spaces, weighted spaces, Lipschitz–Hölder spaces and rectangular $\operatorname{BMO}$ of several variables.