Let G be a group of automorphisms of a tree X (with set of vertices S) and H a kernel on S × S invariant under the action of G. We want to give an estimate of the $l^r$-operator norm (1 ≤ r ≤ 2) of the operator associated to H in terms of a norm for H. This was obtained by U. Haagerup when G is the free group acting simply transitively on a homogeneous tree. Our result is valid when X is a locally finite tree and one of the orbits of G is the set of vertices at even distance from a given vertex; a technical...

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.