Let $ℬ\left(\mathscr{H}\right)$ be the set of all bounded linear operators acting in Hilbert space $\mathscr{H}$ and ${ℬ}^{+}\left(\mathscr{H}\right)$ the set of all positive selfadjoint elements of $ℬ\left(\mathscr{H}\right)$. The aim of this paper is to prove that for every finite sequence ${\left(A_i\right)}_{i=1}^n$ of selfadjoint, commuting elements of ${ℬ}^{+}\left(\mathscr{H}\right)$ and every natural number $p\ge 1$, the inequality $$\frac{e^p}{p^p}\left(\sum _{i=1}^n{A}_i^p\right)\le exp\left(\sum _{i=1}^n{A}_i\right),$$ holds.

For commuting elements x, y of a unital Banach algebra ℬ it is clear that $\parallel e^{x+y}\parallel \le \parallel e^x\parallel \parallel e^y\parallel$. On the order hand, M. Taylor has shown that this inequality remains valid for a self-adjoint operator x and a skew-adjoint operator y, without the assumption that they commute. In this paper we obtain similar inequalities under conditions that lie between these extremes. The inequalities are used to deduce growth estimates of the form $\parallel e^{\text{'}}\parallel \le c(1+|\xi |{⟩}^s$ for all $\xi \in R^m$, where $x=(x_1,...,x_m)\in {ℬ}^m$ and c, s are constants.

We say that the function $f:[a,b]\to ℝ$ is under the chord if $$\frac{\left(b-t\right)f\left(a\right)+\left(t-a\right)f\left(b\right)}{b-a}\ge f\left(t\right)$$ for any $t\in [a,b]$. In this paper we proved amongst other that $${\int }_a^{b}u\left(t\right)df\left(t\right)\ge \frac{f\left(b\right)-f\left(a\right)}{b-a}{\int }_a^{b}u\left(t\right)dt$$ provided that $u:[a,b]\to ℝ$ is monotonic nondecreasing and $f:[a,b]\to ℝ$ is continuous and under the chord. Some particular cases for the weighted integrals in connection with the Fejér inequalities are provided. Applications for continuous functions of selfadjoint operators on Hilbert spaces are also given.

We obtain several characterizations for the classes of Riesz and inessential operators, and apply them to extend the family of Banach spaces for which the essential incomparability class is known, solving partially a problem posed in [6].

Inner-outer factorization for matrix-valued functions defined on totally ordered groups has been considered by Helson and Lowdenslager in connection with multivariate prediction theory. We discuss their result in an operator-theoretic framework and prove that there are obstructions to its extension to operator-valued functions.

In this paper integral formulae, based on Taylor's functional calculus for several operators, are found. Special cases of these formulae include those of Vasilescu and Janas, and an integral formula for commuting operators with real spectra.

In this paper, we give an integral representation for the boundary values of derivatives of functions of the de Branges–Rovnyak spaces $\mathscr{H}\left(b\right)$, where $b$ is in the unit ball of $H^{\infty }\left({ℂ}_{+}\right)$. In particular, we generalize a result of Ahern–Clark obtained for functions of the model spaces $K_b$, where $b$ is an inner function. Using hypergeometric series, we obtain a nontrivial formula of combinatorics for sums of binomial coefficients. Then we apply this formula to show the norm convergence of reproducing kernel $k_{\omega ,n}^b$ of evaluation...