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Register a new userSome Inequalities for Continuous Algebra-Multiplications on a Banach Space
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Maysam Maysami Sadr
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In this short note, we first consider some inequalities for comparison of some algebraic properties of two continuous algebra-multiplications on an arbitrary Banach space and then, as an application, we consider some very basic observations on the space of all continuous algebra-multiplications on a Banach space.
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We characterize strong continuity of general operator semigroups on some Lebesgue spaces. In particular, a characterization of strong continuity of weighted composition semigroups on classical Hardy spaces and weighted Bergman spaces with regular weights is given. As applications, our result improves the results of Siskakis, A. G. cite{AG1} and K{o}nig, W. cite{K} and answers a question of Siskakis, A. G. proposed in cite{AG4}. We also characterize strongly continuous semigroups of weighted composition operators on weighted Bergman spaces in terms of abelian intertwiners of multiplication operator $M_z$.
Let $A$ be a real commutative Banach algebra with unity. Let $a_0in Asetminus{0}$. Let $mathbb Z a_0:={na_0}_{nin mathbb Z}$. Then, $mathbb Z a_0$ is a discrete subgroup of $A$. For any $nin mathbb Z$, the Frechet derivative of the mapping $$x , in , A mapsto x+na_0 , in , A$$ is the identity map on $A$ and, especially, an $A$-linear transformation on $A$. So, the quotient group $A/(mathbb Z a_0)$ is a $1$-dimensional $A$-manifold and the covering projection $$x , in , A mapsto x+mathbb Z a_0 , in , A/(mathbb Z a_0)$$ is an $A$-map. We call $A/(mathbb Z a_0)$ the $1$-dimensional $A$-cylinder by $a_0$. Let $T$ be a compact Hausdorff space. Suppose that there exist $t_1in T$ and $t_2in T$ such that $t_1 ot=t_2$ holds. Then, the set $C(T;mathbb R)$ of all real-valued continuous functions on $T$ is a real commutative Banach algebra with unity and $mathbb R , subsetneq , C(T;mathbb R)$ holds. In this paper, we show that there exists $a_0 , in , C(T;mathbb R)setminus mathbb R$ such that for any $k, in , mathbb N$, the $1$-dimensional $C(T;mathbb R)$-cylinder $(C(T;mathbb R))/(mathbb Z a_0)$ by $a_0$ cannot be embedded in the finite direct product space $(C(T;mathbb R))^k$ as a $C(T;mathbb R)$-submanifold.
We survey some classical norm inequalities of Hardy, Kallman, Kato, Kolmogorov, Landau, Littlewood, and Rota of the type [ |A f|_{mathcal{X}}^2 leq C |f|_{mathcal{X}} big|A^2 fbig|_{mathcal{X}}, quad f in dombig(A^2big), ] and recall that under exceedingly stronger hypotheses on the operator $A$ and/or the Banach space $mathcal{X}$, the optimal constant $C$ in these inequalities diminishes from $4$ (e.g., when $A$ is the generator of a $C_0$ contraction semigroup on a Banach space $mathcal{X}$) all the way down to $1$ (e.g., when $A$ is a symmetric operator on a Hilbert space $mathcal{H}$). We also survey some results in connection with an extension of the Hardy-Littlewood inequality involving quadratic forms as initiated by Everitt.
We consider the inequalities of Gagliardo-Nirenberg and Sobolev in R^d, formulated in terms of the Laplacian Delta and of the fractional powers D^n := (-Delta)^(n/2) with real n >= 0; we review known facts and present novel results in this area. After illustrating the equivalence between these two inequalities and the relations between the corresponding sharp constants and maximizers, we focus the attention on the L^2 case where, for all sufficiently regular f : R^d -> C, the norm || D^j f||_{L^r} is bounded in terms of || f ||_{L^2} and || D^n f ||_{L^2} for 1/r = 1/2 - (theta n - j)/d, and suitable values of j,n,theta (with j,n possibly noninteger). In the special cases theta = 1 and theta = j/n + d/2 n (i.e., r = + infinity), related to previous results of Lieb and Ilyin, the sharp constants and the maximizers can be found explicitly; we point out that the maximizers can be expressed in terms of hypergeometric, Fox and Meijer functions. For the general L^2 case, we present two kinds of upper bounds on the sharp constants: the first kind is suggested by the literature, the second one is an alternative proposal of ours, often more precise than the first one. We also derive two kinds of lower bounds. Combining all the available upper and lower bounds, the Gagliardo-Nirenberg and Sobolev sharp constants are confined to quite narrow intervals. Several examples are given.
We study some fundamental properties of semicocycles over semigroups of self-mappings of a domain in a Banach space. We prove that any semicocycle over a jointly continuous semigroup is itself jointly continuous. For semicocycles over semigroups which have generator, we establish a sufficient condition for differentiablity with respect to the time variable, and hence for the semicocycle to satisfy a linear evolution problem, giving rise to the notion of `generator of a semicocycle. Bounds on the growth of a semicocycle with respect to the time variable are given in terms of this generator. Special consideration is given to the case of holomorphic semicocycles, for which we prove an exact correspondence between certain uniform continuity properties of a semicocyle and boundedness properties of its generator.
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