We investigate the translation lengths of group elements that arise in random walks on the isometry groups of Gromov hyperbolic spaces. In particular, without any moment condition, we prove that non-elementary random walks exhibit at least linear growth of translation lengths. As a corollary, almost every random walk on mapping class groups eventually becomes pseudo-Anosov, and almost every random walk on $\mathrm {Out}(F_n)$ eventually becomes fully irreducible. If the underlying measure further has finite first moment, then the growth rate of translation lengths is equal to the drift, the escape rate of the random walk.

We then apply our technique to investigate the random walks induced by the action of mapping class groups on Teichmüller spaces. In particular, we prove the spectral theorem under finite first moment condition, generalizing a result of Dahmani and Horbez.

Let $T$ be a quadratic operator on a complex Hilbert space $H$ . We show that $T$ can be written as a product of two positive contractions if and only if $T$ is of the form

$$aI\,\oplus \,bI\,\oplus \left( \begin{matrix} aI & P \\ 0 & bI \\ \end{matrix} \right)\,\text{on}\,{{H}_{1}}\,\oplus \,{{H}_{2}}\,\oplus \,\left( {{H}_{3\,}}\,\oplus \,{{H}_{3}} \right)$$

for some $a,\,b\,\in \,\left[ 0,\,1 \right]$ and strictly positive operator $P$ with $\left\| P \right\|\,\le \,\left| \sqrt{a}-\sqrt{b} \right|\sqrt{\left( 1-a \right)\left( 1-b \right)}$ . Also, we give a necessary condition for a bounded linear operator $T$ with operator matrix $\left( \begin{matrix} {{T}_{1}} & {{T}_{3}} \\ 0 & {{T}_{2}} \\ \end{matrix} \right)$ on $H\,\oplus \,K$ that can be written as a product of two positive contractions.

We show that the following results in the classical theory of unbounded linear operators on Hilbert spaces can be proved within the framework of Bishop's constructive mathematics: the Kato-Rellich theorem, the spectral theorem. Stone's theorem, and the self-adjointness of the most common quantum mechanical operators, including the Hamiltonians of electro-magnetic fields with some general forms of potentials.

We construct a suitable representation of a C*-algebra that carries single elements to rank one operators. We also prove an abstract spectral theorem for compact elements in the algebra. This leads naturally to an abstract definition of Cp-classes of compact elements in the algebra.