Some Ostrowski’s type inequalities for the Riemann-Stieltjes integral ${\int }_a^{b}f\left(e^{it}\right)du\left(t\right)$ of continuous complex valued integrands $f:𝒞\left(0,1\right)\to ℂ$ defined on the complex unit circle $𝒞\left(0,1\right)$ and various subclasses of integrators $u:\left[a,b\right]\subseteq \left[0,2\pi \right]\to ℂ$ of bounded variation are given. Natural applications for functions of unitary operators in Hilbert spaces are provided as well.

An exact criterion is derived for an operator valued weight function $W(e^{is},e^{it})$ on the torus to have a factorization $W(e^{is},e^{it})=\Phi (e^{is},e^{it})*\Phi (e^{is},e^{it})$, where the operator valued Fourier coefficients of Φ vanish outside of the Helson-Lowdenslager halfplane $\Lambda =(m,n)\in {\mathbb{Z}}^2:m\ge 1\cup (0,n):n\ge 0$, and Φ is “outer” in a related sense. The criterion is expressed in terms of a regularity condition on the weighted space $L^2\left(W\right)$ of vector valued functions on the torus. A logarithmic integrability test is also provided. The factor Φ is explicitly constructed in terms of Toeplitz operators...