The Hilbert matrix acts on Hardy spaces by multiplication with Taylor coefficients. We find an upper bound for the norm of the induced operator.

On montre que si $T$ est une contraction à spectre dénombrable et telle que, pour tout $ϵ\>0$$$\parallel T^{-1}...$$

In this paper the notion of the derivative of the norm of a linear mapping in a normed vector space is introduced. The fundamental properties of the derivative of the norm are established. Using these properties, linear differential equations in a Banach space are studied and lower and upper estimates of the norms of their solutions are derived.

We discuss boundedness and compactness properties of the embedding $M_{\Lambda }^1\subset L^1\left(\mu \right)$, where $M_{\Lambda }^1$ is the closed linear span of the monomials $x^{{\lambda }_n}$ in $L^1\left([0,1]\right)$ and $\mu$ is a finite positive Borel measure on the interval $[0,1]$. In particular, we introduce a class of “sublinear” measures and provide a rather complete solution of the embedding problem for the class of quasilacunary sequences $\Lambda$. Finally, we show how one can recapture some of Al Alam’s results on boundedness and the essential norm of weighted composition operators from $M_{\Lambda }^1$...

Let $K\subset {ℝ}^m$ ($m\ge 2$) be a compact set; assume that each ball centered on the boundary $B$ of $K$ meets $K$ in a set of positive Lebesgue measure. Let $C_0^{\left(1\right)}$ be the class of all continuously differentiable real-valued functions with compact support in ${ℝ}^m$ and denote by ${\sigma }_m$ the area of the unit sphere in ${ℝ}^m$. With each $\varphi \in C_0^{\left(1\right)}$ we associate the function $$W_K\varphi \left(z\right)=\frac{1}{{\sigma }_m}\underset{{ℝ}^m\setminus K}{\to }\int \mathrm{g}rad\varphi \left(x\right)·\frac{z-x}{{|z-x|}^m}x$$ of the variable $z\in K$ (which is continuous in $K$ and harmonic in $K\setminus B$). $W_K\varphi$ depends only on the restriction ${\varphi |}_B$ of $\varphi$ to the boundary $B$ of $K$. This gives rise to a linear operator $W_K$ acting from...

We characterize all the extreme points of the unit ball in the space of trilinear forms on the Hilbert space ${ℂ}^2$. This answers a question posed by R. Grząślewicz and K. John [7], who solved the corresponding problem for the real Hilbert space ${ℝ}^2$. As an application we determine the best constant in the inequality between the Hilbert-Schmidt norm and the norm of trilinear forms.

The formula $\varrho \left(M\right)=max{\varrho }_{\chi }\left(M\right),r\left(M\right)$ is proved for precompact sets M of weakly compact operators on a Banach space. Here ϱ(M) is the joint spectral radius (the Rota-Strang radius), ${\varrho }_{\chi }\left(M\right)$ is the Hausdorff spectral radius (connected with the Hausdorff measure of noncompactness) and r(M) is the Berger-Wang radius.

Let X be a quasi-Banach rearrangement invariant space and let T be an (ε,δ)-atomic operator for which a restricted type estimate of the form $\parallel T{\chi }_E{\parallel }_X\le D\left(\right|E\left|\right)$ for some positive function D and every measurable set E is known. We show that this estimate can be extended to the set of all positive functions f ∈ L¹ such that ${\left|\right|f\left|\right|}_{\infty }\le 1$, in the sense that $\parallel Tf{\parallel }_X\le D\left(\right|\left|f\right|\left|₁\right)$. This inequality allows us to obtain strong type estimates for T on several classes of spaces as soon as some information about the galb of the space X is known. In this paper...

Let A ∈ B(H) and B ∈ B(K). We say that A and B satisfy the Fuglede-Putnam theorem if AX = XB for some X ∈ B(K,H) implies A*X = XB*. Patel et al. (2006) showed that the Fuglede-Putnam theorem holds for class A(s,t) operators with s + t < 1 and they mentioned that the case s = t = 1 is still an open problem. In the present article we give a partial positive answer to this problem. We show that if A ∈ B(H) is a class A operator with reducing kernel and B* ∈ B(K) is a class 𝓨 operator, and AX =...

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...