### $\langle 2,1\rangle $-compact operators.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Let $X$ be a quasicomplete locally convex Hausdorff space. Let $T$ be a locally compact Hausdorff space and let ${C}_{0}\left(T\right)=\{f\phantom{\rule{0.222222em}{0ex}}T\to I$, $f$ is continuous and vanishes at infinity$\}$ be endowed with the supremum norm. Starting with the Borel extension theorem for $X$-valued $\sigma $-additive Baire measures on $T$, an alternative proof is given to obtain all the characterizations given in [13] for a continuous linear map $u\phantom{\rule{0.222222em}{0ex}}{C}_{0}\left(T\right)\to X$ to be weakly compact.

This article provided some sufficient or necessary conditions for a class of integral operators to be bounded on mixed norm spaces in the unit ball.

In [P] we characterize the pairs of weights for which the fractional integral operator ${I}_{\gamma}$ of order $\gamma $ from a weighted Lebesgue space into a suitable weighted $BMO$ and Lipschitz integral space is bounded. In this paper we consider other weighted Lipschitz integral spaces that contain those defined in [P], and we obtain results on pairs of weights related to the boundedness of ${I}_{\gamma}$ acting from weighted Lebesgue spaces into these spaces. Also, we study the properties of those classes of weights and compare...

The purpose of this note is to give an explicit construction of a bounded operator T acting on the space L2[0,1] such that |Tf(x)| ≤ ∫01 |f(y)| dy for a.e. x ∈ [0.1], and, nevertheless, ||T||Sp = ∞ for every p < 2. Here || ||Sp denotes the norm associated to the Schatten-Von Neumann classes.

Analytic extensions of the metaplectic representation by integral operators of Gaussian type have been calculated in the ${L}^{2}({\mathbb{R}}^{n})$ and the Bargmann-Fock realisations by Howe [How2] and Brunet-Kramer [Brunet-Kramer, Reports on Math. Phys., 17 (1980), 205-215]], respectively. In this paper we show that the resulting semigroups of operators are isomorphic and calculate the intertwining operator.

On a compact metric space X one defines a transition system to be a lower semicontinuous map $X\to {2}^{X}$. It is known that every Markov operator on C(X) induces a transition system on X and that commuting of Markov operators implies commuting of the induced transition systems. We show that even in finite spaces a pair of commuting transition systems may not be induced by commuting Markov operators. The existence of trajectories for a pair of transition systems or Markov operators is also investigated.

In this paper we characterize those bounded linear transformations $Tf$ carrying ${L}^{1}\left({\mathbb{R}}^{1}\right)$ into the space of bounded continuous functions on ${\mathbb{R}}^{1}$, for which the convolution identity $T(f*g)=Tf\xb7Tg$ holds. It is shown that such a transformation is just the Fourier transform combined with an appropriate change of variable.