Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0903.2038

Properties of Representations of Operators acting between Spaces of Vector-Valued Functions

Delio Mugnolo·Robin Nittka
View original source ↗Read source document ↗Export metadata ↓

Abstract

A well-known result going back to the 1930s states that all bounded linear operators mapping scalar-valued $L^1$-spaces into $L^\infty$-spaces are kernel operators and that in fact this relation induces an isometric isomorphism between the space of such operators and the space of all bounded kernels. We extend this result to the case of spaces of vector-valued functions. A recent result due to Arendt and Thomaschewski states that the local operators acting on $L^p$-spaces of functions with values in separable Banach spaces are precisely the multiplication operators. We extend this result to non-separable dual spaces. Moreover, we relate positivity and other order properties of the operators to corresponding properties of the representations.

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Delio Mugnolo, Robin Nittka. 2009-03-12. Properties of Representations of Operators acting between Spaces of Vector-Valued Functions. https://doi.org/10.1007/s11117-010-0045-0

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Spaces with the maximal projection constant revisited

Let $n \geq 2$ be an integer such that an equiangular set of vectors $w_1, \ldots, w_d$ of the maximal possible cardinality (that is, attaining the classical Gerzon upper bound) exists in $\mathbb{K}^n$, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ (so that $d=\frac{n(n+1)}{2}$ in the real case and $d=n^2$ in the complex case). We provide a complete characterization of $n$-dimensional normed spaces whose absolute projection constant is maximal among all $n$-dimensional normed spaces over $\mathbb{K}$. The characterization states that $X$ has the maximal projection constant if and only if it is isometric to a space whose dual unit ball is contained between the absolutely convex hull of the vectors $w_1, \ldots, w_d$ and a suitably rescaled zonotope generated by the same vectors. As a consequence, we obtain that, in the considered situations, $n=2$ with $\mathbb{K}=\mathbb{R}$ is the only case in which there is, up to isometry, a unique norm on $\mathbb{K}^n$ with the maximal projection constant. In this case, the unit ball is a linear image of a regular hexagon in $\mathbb{R}^2$.

math.FA↗
arXiv2026

Subdyadic time-frequency analysis: Gabor frames, modulation spaces, and Miyachi multipliers

We present a time-frequency framework adapted to dispersive phase functions via a subdyadic geometry in phase space. On top of this geometry we construct stable frequency-adaptive Gabor-type frames with quantitative control of overlap, almost orthogonality, and off-diagonal decay. Based on these frames we introduce modulation spaces consistent with the subdyadic scale and establish window and lattice independence, identifications in the Hilbertian case, duality, and natural inclusion relations. Within this setting we study high-frequency H"ormander--Miyachi multipliers, relying on discrete block almost diagonalization and direct localization estimates for the primal and canonical dual frames, and obtain boundedness on weighted modulation spaces. Finally, we give a subdyadic Gabor-frame characterization of H"ormander's classical local wavefront set and recover the standard microlocality and ellipticity properties of order-zero pseudodifferential operators. Taken together, these results provide a unified analytical framework for time--frequency analysis, dispersive multiplier theory, and local microlocal analysis in the subdyadic geometry.

math.FA↗
arXiv2026

B-Frames, B-Riesz bases, and Their tensor products

Like g-frames, b-frames were introduced to generalize the concept of frames, allowing for broader applications in signal processing and other fields. The advantage of b-frames resides in their simpler definition, which may lead to reduced processing times. In this paper, we define dual b-frames and b-Riesz bases which were not precisely defined in previous literature and provide several characterizations of b-Riesz bases. We prove that the tensor product of two sequences, each lying in a Hilbert space, constitutes a b-frame (or a b-Riesz basis) if and only if both components of the product are b-frames (or b-Riesz bases). Finally, we establish a correspondence between b-frames and g-frames and propose a process for constructing frames induced by b-frames.

math.FA↗