2000 Mathematics Subject Classification: Primary: 47B47, 47B10; secondary 47A30.Let H be an infinite-dimensional complex Hilbert space and let A, B ∈ L(H), where L(H) is the algebra of operators on H into itself. Let δAB: L(H) → L(H) denote the generalized derivation δAB(X) = AX − XB. This note will initiate a study on the class of pairs (A,B) such that [‾(R(δAB))] = [‾(R(δB*A*))]; i.e. [‾(R(δAB))] is self-adjoint.

Let $\parallel ·\parallel$ be a norm on the algebra ${ℳ}_n$ of all $n\times n$ matrices over $ℂ$. An interesting problem in matrix theory is that “Are there two norms ${\parallel ·\parallel }_1$ and ${\parallel ·\parallel }_2$ on ${ℂ}^n$ such that $\parallel A\parallel =max\{\parallel Ax{\parallel }_2\parallel x{\parallel }_1=1\}$ for all $A\in {ℳ}_n$?” We will investigate this problem and its various aspects and will discuss some conditions under which ${\parallel ·\parallel }_1={\parallel ·\parallel }_2$.

Several authors have defined operational quantities derived from the norm of an operator between Banach spaces. This situation is generalized in this paper and we present a general framework in which we derivate several maps $X\to ℝ$ from an initial one $X\to ℝ$, where $X$ is a set endowed with two orders, $\le$ and ${\le }^{*}$, related by certain conditions. We obtain only three different derivated maps, if the initial map is bounded and monotone.

We characterize the UMD-property of a Banach space X by sequences of ideal norms associated with trigonometric orthonormal systems. The asymptotic behavior of those numerical parameters can be used to decide whether X is a UMD-space. Moreover, if this is not the case, we obtain a measure that shows how far X is from being a UMD-space. The main result is that all described sequences are not only simultaneously bounded but are also asymptotically equivalent.

We denote by the unit circle and by the unit disc of ℂ. Let s be a non-negative real and ω a weight such that $\omega \left(n\right)={(1+n)}^s$ (n ≥ 0) and the sequence ${(\omega (-n)/{(1+n)}^s)}_{n\ge 0}$ is non-decreasing. We define the Banach algebra $A_{\omega }\left(\right)={f\in \left(\right):\left|\right|f\left|\right|}_{\omega }={\sum }_{n=-\infty }^{+\infty }\left|f̂\left(n\right)\right|\omega \left(n\right)<+\infty$. If I is a closed ideal of $A_{\omega }\left(\right)$, we set $h⁰\left(I\right)=z\in :f\left(z\right)=0\left(f\in I\right)$. We describe all closed ideals I of $A_{\omega }\left(\right)$ such that h⁰(I) is at most countable. A similar result is obtained for closed ideals of the algebra $A{⁺}_s\left(\right)=f\in A_{\omega }\left(\right):f̂\left(n\right)=0(n<0)$ without inner factor. Then we use this description to establish a link between operators with countable spectrum and interpolating sets...

By virtue of convexity of Heinz means, in this paper we derive several refinements of Heinz norm inequalities with the help of the Jensen functional and its properties. In addition, we discuss another approach to Heinz operator means which is more convenient for obtaining the corresponding operator inequalities for positive invertible operators.

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.

The behavior of the essential spectrum and the essential norm under (complex/real) interpolation is investigated. We extend an example of Albrecht and Müller for the spectrum by showing that in complex interpolation the essential spectrum ${\sigma }_e\left(S_{\left[\theta \right]}\right)$ of an interpolated operator is also in general a discontinuous map of the parameter θ. We discuss the logarithmic convexity (up to a multiplicative constant) of the essential norm under real interpolation, and show that this holds provided certain compact approximation...

Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce a new class, denoted *, of operators satisfying $T^{*k}\left(\right|T²|-|T*|²)T^k\ge 0$ where k is a natural number, and we prove basic structural properties of these operators. Using these results, we also show that if E is the Riesz idempotent for a non-zero isolated point μ of the spectrum of T ∈ *, then E is self-adjoint and EH = ker(T-μ) = ker(T-μ)*. Some spectral properties are also presented.