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 =...