Let G be a compact and connected semisimple Lie group and Σ an invariant control systems on G. Our aim in this work is to give a new proof of Theorem 1 proved by Jurdjevic and Sussmann in \cite{ju-su 2}. Precisely, to find a positive time sΣ such that the system turns out controllable at uniform time sΣ. Our proof is different, elementary and the main argument comes directly from the definition of semisimple Lie group. The uniform time is not arbitrary. Finally, if A=⋂t>0A(t,e) denotes the reachable set from arbitrary uniform time, we conjecture that it is possible to determine A as the intersection of the isotropy groups of orbits of G-representations which contains exp(z), where z is the Lie algebra determined by the control vectors.
In this paper, we limit our analysis to the difference of the weighted composition operators acting from the Hardy space to weighted-type space in the unit ball of $\mathbb {C}^N$, and give some necessary and sufficient conditions for their boundedness or compactness. The results generalize the corresponding results on the single weighted composition operators and on the differences of composition operators, for example, M. Lindström and E. Wolf: Essential norm of the difference of weighted composition operators. Monatsh. Math. 153 (2008), 133-143.
We apply the general theory of $\tau $-Corson Compact spaces to remove an unnecessary hypothesis of zero-dimensionality from a theorem on polyadic spaces of tightness $\tau $. In particular, we prove that polyadic spaces of countable tightness are Uniform Eberlein compact spaces.
In this paper, we study the monotone meta-Lindelöf property. Relationships between monotone meta-Lindelöf spaces and other spaces are investigated. Behaviors of monotone meta-Lindelöf $GO$-spaces in their linearly ordered extensions are revealed.
Let P be a topological property. A space X is said to be star P if whenever U is an open cover of X, there exists a subspace A ⊆ X with property P such that X = St(A,U), where St(A, U) = ∪ {U ∈ U : U ∩A ≠ ∅}. In this paper, we study the relationships of star P properties for P ∈ {Lindelöf, compact, countably compact} in pseudocompact spaces by giving some examples.
Let $X$ be a Banach space. We give characterizations of when ${\cal F}(Y,X)$ is a $u$-ideal in ${\cal W}(Y,X)$ for every Banach space $Y$ in terms of nets of finite rank operators approximating weakly compact operators. Similar characterizations are given for the cases when ${\cal F}(X,Y)$ is a $u$-ideal in ${\cal W}(X,Y)$ for every Banach space $Y$, when ${\cal F}(Y,X)$ is a $u$-ideal in ${\cal W}(Y,X^{**})$ for every Banach space $Y$, and when ${\cal F}(Y,X)$ is a $u$-ideal in ${\cal K}(Y,X^{**})$ for every Banach space $Y$.