We construct two Banach algebras, one which contains analytic semigroups ${\left(a^z\right)}_{Rez>0}$ such that $|a^{1+iy}|\to \infty$ arbitrarily slowly as $\left|y\right|\to \infty$, the other which contains ones such that $|a^{1+iy}|\to \infty$ arbitrarily fast

Let X be a completely regular topological space and A a commutative locally m-convex algebra. We give a description of all closed and in particular closed maximal ideals of the algebra C(X,A) (= all continuous A-valued functions defined on X). The topology on C(X,A) is defined by a certain family of seminorms. The compact-open topology of C(X,A) is a special case of this topology.

We prove the ${}_p$-spectral radius formula for n-tuples of commuting Banach algebra elements

In a recent paper by H. X. Cao, J. H. Zhang and Z. B. Xu an $\alpha$-Lipschitz operator from a compact metric space into a Banach space $A$ is defined and characterized in a natural way in the sence that $F:K\to A$ is a $\alpha$-Lipschitz operator if and only if for each $\sigma \in X^{*}$ the mapping $\sigma \circ F$ is a $\alpha$-Lipschitz function. The Lipschitz operators algebras $L^{\alpha }(K,A)$ and $l^{\alpha }(K,A)$ are developed here further, and we study their amenability and weak amenability of these algebras. Moreover, we prove an interesting result that $L^{\alpha }(K,A)$ and $l^{\alpha }(K,A)$ are isometrically...

We consider the Fejér (or first arithmetic) means of double Fourier series of functions belonging to one of the Hardy spaces $H^{(1,0)}{(}^2)$, $H^{(0,1)}{(}^2)$, or $H^{(1,1)}{(}^2)$. We prove that the maximal Fejér operator is bounded from $H^{(1,0)}{(}^2)$ or $H^{(0,1)}{(}^2)$ into weak-$L^1{(}^2)$, and also bounded from $H^{(1,1)}{(}^2)$ into $L^1{(}^2)$. These results extend those by Jessen, Marcinkiewicz, and Zygmund, which involve the function spaces $L^{1}lo{g}^{+}L{(}^2)$, $L^1{\left(lo{g}^{+}L\right)}^2{(}^2)$, and $L^{\mu }{(}^2)$ with 0 < μ < 1, respectively. We establish analogous results for the maximal conjugate Fejér operators. On closing, we formulate two conjectures....

Let C(Ω) be the algebra of all complex-valued continuous functions on a topological space Ω where C(Ω) contains unbounded functions. First it is shown that C(Ω) cannot have a Banach algebra norm. Then it is shown that, for certain Ω, C(Ω) cannot possess an (incomplete) normed algebra norm. In particular, this is so for $\Omega ={ℝ}^n$ where ℝ is the reals.

We prove that, for certain domains $\mathbb{D}$, continuous product of domains $D_{\omega }$, the Carathéodory pseudodistance satisfies the following product property $C_{\mathbb{D}}\left(f,g\right)={sup}_{\omega }{C}_{D_{\omega }}\left(f\left(\omega \right),g\left(\omega \right)\right)$