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F. Sukochev

Publications and source records attributed to F. Sukochev.

At least 19 recordsLinked to original sources
arXiv2026

Schur Multipliers with Unequal Operator and Completely Bounded Norms on $S_p$, $1<p\ne 2<\infty $

For every $1<p\neq2<\infty$, we exhibit an explicit Schur multiplier on $S_p$ whose operator norm is strictly smaller than its completely bounded norm. More precisely, for each such $p$, we construct a finitely supported Schur symbol $m_p$ such that \[ \|M_{m_p}\|_{p\to p} < \|M_{m_p}\|_{\mathrm{cb},p}. \] This answers the question raised by Lafforgue and de la Salle in 2011 after their Conjecture~1.8 and gives an affirmative answer to Statement~2 in Section~5 of Caspers and Wildschut (2019). In particular, it disproves Statement~3 of Caspers and Wildschut (2019) throughout the same range of exponents.

math.OA↗
arXiv2026

Trigonometric bases in noncommutative $L_p(\mathbb{T}^d_\theta)$ spaces and associated partial sum operators

We develop a harmonic-analytic method for constructing a generalized trigonometric system in noncommutative $L_p(\mathbb{T}^d_\theta)$ spaces arising from the strongly continuous representation of $\mathbb{T}^d$ and show that the generalized trigonometric system is a Schauder basis in $L_p(\mathbb{T}^d_\theta)$ for $1<p<\infty.$ In particular, we prove that this trigonometric system forms an RUC-basis in $L_p(\mathbb{T}^d_\theta)$ for $2<p<\infty.$ Our results provide a noncommutative counterpart of the classical trigonometric basis in $L_p(\mathbb{T}^d)$. Further, we obtain a weak $(1,1)$ type estimate of partial sum operators associated with noncommutative trigonometric systems. This allows us to study uniformly boundedness of partial sum operators between pairs of symmetric spaces that do not necessarily possess nontrivial Boyd indices, extending known results in this direction to the setting of quasi-Banach symmetric spaces.

math.OA↗
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↗
arXiv2020

Extreme points of the set of elements majorised by an integrable function: Resolution of a problem by Luxemburg and of its noncommutative counterpart

Let $f$ be an arbitrary integrable function on a finite measure space $(X,Σ, ν)$. We characterise the extreme points of the set $Ω(f)$ of all measurable functions on $(X,Σ, ν)$ majorised by $f$, providing a complete answer to a problem raised by W.A.J. Luxemburg in 1967. Moreover, we obtain a noncommutative version of this result.

math.FA↗
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↗
arXiv2019

Interpolation between $L_0({\mathcal M},τ)$ and $L_\infty({\mathcal M},τ)$

Let ${\mathcal M}$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $τ$. We show that the symmetrically $Δ$-normed operator space $E({\mathcal M},τ)$ corresponding to an arbitrary symmetrically $Δ$-normed function space $E(0,\infty)$ is an interpolation space between $L_0({\mathcal M},τ)$ and ${\mathcal M}$, which is in contrast with the classical result that there exist symmetric operator spaces $E({\mathcal M},τ)$ which are not interpolation spaces between $L_1({\mathcal M},τ)$ and ${\mathcal M}$. Besides, we show that the ${\mathcal K}$-functional of every $X\in L_0({\mathcal M},τ)+ {\mathcal M} $ coincides with the ${\mathcal K}$-functional of its generalized singular value function $μ(X)$. Several applications are given, e.g., it is shown that the pair $(L_0({\mathcal M},τ),{\mathcal M})$ is ${\mathcal K}$-monotone when ${\mathcal M}$ is a non-atomic finite factor.

math.OA↗
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

Spectral flow for nonunital spectral triples

We prove two results about nonunital index theory left open by [CGRS2]. The first is that the spectral triple arising from an action of the reals on a C*-algebra with invariant trace satisfies the hypotheses of the nonunital local index formula. The second result concerns the meaning of spectral flow in the nonunital case. For the special case of paths arising from the odd index pairing for smooth spectral triples in the nonunital setting we are able to connect with earlier approaches to the analytic definition of spectral flow.

math.KT↗