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D. Zanin

Publications and source records attributed to D. Zanin.

At least 19 recordsLinked to original sources
arXiv2026

Fractional Sobolev embeddings on noncommutative torus

In this paper, we study the noncommutative fractional symmetric Sobolev spaces on noncommutative torus. We prove noncommutative distributional fractional Sobolev inequality and as its application, we obtain Sobolev embeddings. In order to obtain these results, we first prove a noncommutative version of the famous O'Neil inequality for the convolution. As a first application of our main results, we obtain a Cwikel-Solomyak-type estimate. As an another application, we show a $L_2$-time decay for the mild solution of the Cauchy problem for the diffusion equation in this noncommutative setting. When $\theta=0,$ our results recover many known results on Sobolev embedding on the torus.

math.AP↗
arXiv2026

Optimal distributional estimates of the multiple Hilbert transform

In this paper, we study optimal distributional estimates for the multiple Hilbert transform. We obtain pointwise upper and lower distributional estimates of the multiple Hilbert transform in terms of the $d$-fold composition of the Calder\'{o}n operator with itself. This extends the fundamental results by A. P. Calder\'{o}n, D. Boyd, and Ch. Fefferman for arbitrary $d\in\mathbb N.$

math.FA↗
arXiv2023

$C^{\ast}$-algebraic approach to the principal symbol. III

We treat the notion of principal symbol mapping on a compact smooth manifold as a $\ast$-homomorphism of $C^{\ast}$-algebras. Principal symbol mapping is built from the ground, without referring to the pseudodifferential calculus on the manifold. Our concrete approach allows us to extend Connes Trace Theorem for compact Riemannian manifolds.

math.OA↗
arXiv2022

An Application of Singular Traces to Crystals and Percolation

For a certain class of discrete metric spaces, we provide a formula for the density of states. This formula involves Dixmier traces and is proven using recent advances in operator theory. Various examples are given of metric spaces for which this formula holds, including crystals, quasicrystals and the infinite cluster resulting from super-critical bond percolation on $\mathbb{Z}^d$.

math-ph↗
arXiv2019

The boundedness of the Hilbert transformation from one rearrangement invariant Banach space into another and applications

In this paper, we study the boundedness of the Hilbert transformation in Lorentz function spaces, thereby complementing classical results of Boyd. We also characterize the optimal range of a triangular truncation operator in Schatten-Lorentz ideals. These results further entail sharp commutator estimates and applications to operator Lipschitz functions in Schatten-Lorentz ideals.

math.FA↗
arXiv2019

The optimal range of the Calderòn operator and its applications

We identify the optimal range of the Calderòn operator and that of the classical Hilbert transform in the class of symmetric quasi-Banach spaces. Further consequences of our approach concern the optimal range of the triangular truncation operator, operator Lipschitz functions and commutator estimates in ideals of compact operators.

math.FA↗
arXiv2017

Embeddings of operator ideals into $\mathcal{L}_p-$spaces on finite von Neumann algebras

Let $\mathcal{L}(H)$ be the $*$-algebra of all bounded operators on an infinite dimensional Hilbert space $H$ and let $(\mathcal{I}, \|\cdot\|_{\mathcal{I}})$ be an ideal in $\mathcal{L}(H)$ equipped with a Banach norm which is distinct from the Schatten-von Neumann ideal $\mathcal{L}_p(\mathcal{H})$, $1\leq p<2$. We prove that $\mathcal{I}$ isomorphically embeds into an $L_p$-space $\mathcal{L}_p(\mathcal{R}),$ $1\leq p<2,$ (here, $\mathcal{R}$ is the hyperfinite II$_1$-factor) if its commutative core (that is, Calkin space for $\mathcal{I}$) isomorphically embeds into $L_p(0,1).$ Furthermore, we prove that an Orlicz ideal $\mathcal{L}_M(H)\neq\mathcal{L}_p(H)$ isomorphically embeds into $\mathcal{L}_p(\mathcal{R}),$ $1\leq p<2,$ if and only if it is an interpolation space for the Banach couple $(\mathcal{L}_p(H),\mathcal{L}_2(H)).$ Finally, we consider isomorphic embeddings of $(\mathcal{I}, \|\cdot\|_{\mathcal{I}})$ into $L_p$-spaces associated with arbitrary finite von Neumann algebras.

math.OA↗
arXiv2016

Determinants associated to traces on operator bimodules

Given a II$_1$-factor $\mathcal{M}$ with tracial state $τ$ and given an $\mathcal{M}$-bimodule $\mathcal{E}(\mathcal{M},τ)$ of operators affiliated to $\mathcal{M}$ and a trace $φ$ on $\mathcal{E}(\mathcal{M},τ)$, (namely, a linear functional that is invariant under unitary conjugation), we prove that $\det_φ:\mathcal{E}_{\log}(\mathcal{M},τ)\to[0,\infty)$ defined by $\det_φ(T)=\exp(φ(\log |T|))$ is a multiplicative map on the set $\mathcal{E}_{\log}(\mathcal{M},τ)$ of all affiliated operators $T$ such that $\log_+(|T|)\in\mathcal{E}(\mathcal{M},τ)$. Finally, we show that all multiplicative maps on the invertible elements of $\mathcal{E}_{\log}(\mathcal{M},τ)$ arise in this fashion.

math.OA↗
arXiv2014

On uniqueness of distribution of a random variable whose independent copies span a subspace in L_p

Let 1\leq p<2 and let L_p=L_p[0,1] be the classical L_p-space of all (classes of) p-integrable functions on [0,1]. It is known that a sequence of independent copies of a mean zero random variable f from L_p spans in L_p a subspace isomorphic to some Orlicz sequence space l_M. We present precise connections between M and f and establish conditions under which the distribution of a random variable f whose independent copies span l_M in L_p is essentially unique.

math.FA↗
arXiv2014

Weak type estimates for the absolute value mapping

We prove that if A and B are bounded self-adjoint operators such that A-B belongs to the trace class, then |A| -|B| belongs to the principal ideal L_{1,\infty} in the algebra L(H) of all bounded operators on an infinite-dimensional Hilbert space generated by an operator whose sequence of eigenvalues is {1, 1/2, 1/3, 1/4, ...}. Moreover, μ(j;|A| -|B|)\leq const(1 + j)^{-1}\|A-B\|_1. We also obtain a semifinite version of this result, as well as the corresponding commutator estimates.

math.FA↗
arXiv2014

Dixmier traces are weak$^*$ dense in the set of all fully symmetric traces

We extend Dixmier's construction of singular traces (see \cite{Dixmier}) to arbitrary fully symmetric operator ideals. In fact, we show that the set of Dixmier traces is weak$^*$ dense in the set of all fully symmetric traces (that is, those traces which respect Hardy-Littlewood submajorization). Our results complement and extend earlier work of Wodzicki \cite{Wodzicki}.

math.OA↗
arXiv2014

Universal measurability and the Hochschild class of the Chern character

We study notions of measurability for singular traces, and characterise universal measurability for operators in Dixmier ideals. This measurability result is then applied to improve on the various proofs of Connes' identification of the Hochschild class of the Chern character of Dixmier summable spectral triples. The measurability results show that the identification of the Hochschild class is independent of the choice of singular trace. As a corollary we obtain strong information on the asymptotics of the eigenvalues of operators naturally associated to spectral triples (A,H,D) and Hochschild cycles for A.

math.FA↗
arXiv2013

Which traces are spectral?

Among ideals of compact operators on a Hilbert space we identify a subclass of those closed with respect to the logarithmic submajorization. Within this subclass, we answer the questions asked by Pietsch \cite{Pietsch_nachrichten} and by Dykema, Figiel, Weiss and Wodzicki \cite{DFWW}. In the first case, we show that Lidskii-type formulae hold for every trace on such ideal. In the second case, we provide the description of the commutator subspace associated with a given ideal. Finally, we prove that a positive trace on an arbitrary ideal is spectral if and only if it is monotone with respect to the logarithmic submajorization.

math.OA↗
arXiv2013

A decomposition theorem in II_1-factors

Building on results of Haagerup and Schultz, we decompose an arbitrary operator in a diffuse, finite von Neumann algebra into the sum of a normal operator and an s.o.t.-quasinilpotent operator. We also prove an analogue of Weyl's inequality relating eigenvalues and singular values for operators in a diffuse, finite von Neumann algebra.

math.OA↗
arXiv2011

Traces on symmetrically normed operator ideals

For every symmetrically normed ideal $\mathcal{E}$ of compact operators, we give a criterion for the existence of a continuous singular trace on $\mathcal{E}$. We also give a criterion for the existence of a continuous singular trace on $\mathcal{E}$ which respects Hardy-Littlewood majorization. We prove that the class of all continuous singular traces on $\mathcal{E}$ is strictly wider than the class of continuous singular traces which respect Hardy-Littlewood majorization. We establish a canonical bijection between the set of all traces on $\mathcal{E}$ and the set of all symmetric functionals on the corresponding sequence ideal. Similar results are also proved in the setting of semifinite von Neumann algebras.

math.OA↗
arXiv2010

Orbits in symmetric spaces

We characterize those elements in a fully symmetric spaces on the interval $(0,1)$ or on the semi-axis $(0,\infty)$ whose orbits are the norm-closed convex hull of their extreme points. Our results extend and complement earlier work on the same theme by Braverman and Mekler.

math.FA↗