arXiv subjects

Fedor Sukochev

Publications and source records attributed to Fedor Sukochev.

At least 19 recordsLinked to original sources

arXiv2026

Arazy's conjecture concerning Schur multipliers: revisited and resolved

Let $\mathcal{S}^ r$ denote the Schatten--von Neumann class and let $S_{Ψ_{f,λ}}$ be the Schur--Hadamard multiplier whose symbol is the divided-difference matrix of $f$ along $λ$. Let $0<α,r<\infty$, let $f\in C^1([-1,1])$ satisfy $f(0)=0$, $|f'(t)|\lesssim |t|^α$, and let $λ\in\ell^r$ be real with $\|λ\|_{\ell^\infty}\leq1$. We determine the pairs $0<p,q\leq\infty$ for which $S_{Ψ_{f,λ}}:\mathcal{S}^ q\to\mathcal{S}^ p$ is bounded for every such $f$ and $λ$. Our principal new positive estimates treat $q=\infty,1$ and $0<p<1$. Precisely, we show that the boundedness holds exactly when \[ \frac1p\leq\fracα{r} +\min\!\left\{1,\frac1q\right\}. \] In particular, this resolves the untreated cases in [Arazy, PAMS, 1982] and [Potapov, Sukochev, Tomskova, Adv. Math., 2015]. Our method also delivers a new proof of the main results in just cited papers.

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arXiv2026

The UMD property of symmetric operator spaces

We prove that if $E$ is a UMD symmetric Banach function space on $(0,\infty)$, then $E(\mathcal{M},τ)$ is UMD for every semifinite von Neumann algebra $\mathcal{M}$ equipped with a faithful normal semifinite trace $τ$. This resolves in the affirmative an open problem that has circulated in the non-commutative world for more than four decades.

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arXiv2026

The Daugavet property in symmetric operator spaces

Under mild assumptions, the main results of this paper characterize a (real or complex) symmetric operator space affiliated with a semi-finite atomless von Neumann algebra $(\mathcal M, τ)$ possessing the Daugavet property as $L_1(\mathcal M,τ)$ or $\mathcal M$ (up to equivalent norms, or even proportional norms). Our results are new even for complex symmetric function spaces, which extend results of \cite{AKM12, KMMW13, AKM15} concerning the Daugavet property for real symmetric function spaces.

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arXiv2026

Isometries of Haagerup--Schultz algebras

We establish a version of the classical results concerning descriptions of isometries on $C^*$-algebras and noncommutative $L_p$-spaces due to Kadison (1951) and Yeadon (1981) in the setting of Haagerup--Schultz algebras. Precisely, we show that (not necessarily surjective) isometries on such algebras are necessarily implemented by partial isometries and trace-preserving Jordan $^*$-monomorphisms.

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arXiv2026

Norms of multiplication operators: answering Fialkow--Loebl question

Let $\mathcal{M}$ be a factor equipped with a semi-finite faithful normal trace $τ$. Let $E(0,\infty)$ be a symmetrically normed function space and $E(\mathcal{M},τ)$ be the corresponding symmetrically normed operator space. Suppose that $a, b$ are $τ$-measurable operators affiliated with $\mathcal{M}$. It is shown that the range of the multiplication operator $S_{a,b}: x\mapsto axb$ on $\mathcal{M}$ is contained in $E(\mathcal{M}, τ)$ if and only if $μ(a)μ(b)$ belongs to $E(0, \infty)$, where $μ(x)$ stands for the generalized singular value function of a $τ$-measurable operators $x$ affiliated with $\mathcal{M}$. Moreover, we have $$ \|S_{a,b}\|_{\mathcal{M}\to E(\mathcal{M},τ)}= \| μ(a )μ( b) \|_{E (0,\infty) }, $$ which answers a question by Fialkow and Loebl (1984). We also consider the quasi-normed case, and show that the natural quasi-norm of weak $L_p$-space, $0<p<\infty$, is not monotone with respect to the logarithmic submajorisation.

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arXiv2026

Isomorphisms between symmetric spaces over infinite and finite von Neumann algebras

The primary aim of this paper is to show a linear topological isomorphism between (elements of a wide class {of}) symmetric spaces over the hyperfinite $II_1$ factor $\mathcal{R}$ and certain symmetric operator space over the hyperfinite $II_\infty $ factor $\mathcal{R}\bar{\otimes}\mathcal{L}(H))$. Precisely, we show that for any symmetric function space $E(0,1)$ (in the sense of Lindenstrauss and Tzafriri) such that both $E(0,1)$ and its Köthe dual have the Kruglov property, the symmetric operator space $E(\mathcal{R})$ is isomorphic to some symmetric space $Z_E^2(\mathcal{R}\bar{\otimes}\mathcal{L}(H))$. This result establishes a noncommutative version of a well-known result due to Johnson, Maurey, Schechtman and Tzafriri, and answers the noncommutative version of a question due to Mityagin.

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arXiv2026

Weyl asymptotic formulas in the nilpotent Lie group setting

The asymptotic properties of negative order pseudo-differential operators have been an important part of the spectral theory since H.Weyl's classical results. In this paper, we derive a spectral asymptotic formula for the negative fractional powers of hypoelliptic operators on graded Lie groups. Such operators have anisotropically homogeneous principal symbols; for these, our results generalize known results of Birman and Solomyak from 1977. Additionally, our work implies a version of Connes' integration formula for hypoelliptic operators on graded Lie groups. Our methods allow us to extend results from constant-coefficient operators to those with smoothly varying coefficients. The principal technique is to adapt the singular value perturbation arguments of Birman and Solomyak to the setting of nilpotent Lie groups. The decomposing of graded Lie groups is inspired by Folland and Stein in their development of harmonic analysis on homogeneous groups.

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arXiv2026

Schur bounded patterns and submajorisation

We characterise the Schur bounded patterns of ideals of compact operators that are not closed under submajorisation, in particular the Schatten ideals $\mathcal{C}_p$ with $0<p<1.$ Conversely we characterise the ideals that are not closed under submajorisation by their Schur bounded patterns.

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arXiv2026

Arazy-type decomposition theorem for bounded linear operators and commutators on the trace class

The classical Arazy's decomposition theorem provides a powerful tool in the study of sequences in (and isomorphisms on) a separable operator ideal $\mathcal C_E$ of the algebra $\mathcal B(H)$ of all bounded linear operators on the separable infinite-dimensional Hilbert space $H$. In this paper, we extend and strengthen Arazy's decomposition theorem to the setting of general bounded linear operators on a separable (quasi-Banach) operator ideal $\mathcal C_E$ of $\mathcal B(H)$. Several applications are given to the study of $\mathcal C_E$-strictly singular operators, largest proper ideals in the algebra $\mathcal B(\mathcal C_E)$ of all bounded linear operators on $\mathcal C_E$ and complementably homogeneous Banach spaces among others. Our versions of decomposition theorems supply tools for a noncommutative generalization of deep commutator theorems for operators on $\ell_p$ and $L_p$, $1\le p <\infty $, due to Brown and Pearcy, Apostol, and Dosev, Johnson and Schechtman. We are able to characterize commutators on the Schatten-von Neumann class $\mathcal C_p$, $1\le p<\infty $. For the crucial case, $p=1$, we establish that any operator $T\in\mathcal B(\mathcal C_1)$ is a commutator if and only if $T$ is not of the form $λI+K$ for some $λ\neq 0$ and $\mathcal C_1$-strictly singular operator $K$.

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arXiv2025

Isometric Structure in Noncommutative Symmetric Spaces

This is a systematic study of isometries between noncommutative symmetric spaces. Let $\mathcal{M}$ be a semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a separable Hilbert space $\mathcal{H}$ equipped with a semifinite faithful normal trace $τ$. We show that for any noncommutative symmetric space corresponding to a symmetric function space $E(0,\infty)$ in the sense of Lindenstrauss--Tzafriri such that $\left\|\cdot\right\|_E\ne λ\left\|\cdot\right\|_{L_2}$, $λ\in \mathbb{R}_+$, any isometry on $E(\mathcal{M},τ)$ is of elementary form. This answers a long-standing open question raised in the 1980s in the non-separable setting [Math. Z. 1989], while the case of separable symmetric function spaces was treated in [Huang \& Sukochev, JEMS, 2024]. As an application, we obtain a noncommutative Kalton--Randrianantoanina--Zaidenberg Theorem, providing a characterization of noncommutative $L_p$-spaces over finite von Neumann algebras and a necessary and sufficient condition for an operator on a noncommutative symmetric space to be an isometry. Having this at hand, we answer a question posed by Mityagin in 1970 [Uspehi Mat. Nauk] and its noncommutative counterpart by showing the any symmetric space $E(\mathcal{M},τ)\ne L_p(\mathcal{M},τ)$ over a noncommutative probability is not isometric to a symmetric space over a von Neumann algebra equipped with a semifinite infinite faithful normal trace. It is also shown that any noncommutative $L_p$-space, $1\le p<\infty$, affiliated with an atomless semifinite von Neumann algebra has a unique symmetric structure up to isometries. This contributes to the resolution of an isometric version of Pełczyński's problem concerning the uniqueness of the symmetric structure in noncommutative symmetric spaces.

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arXiv2025

Fuglede theorem for symmetric spaces of $τ$-measurable operators

We extend the classical Fuglede commutativity theorem to the full scale of symmetrically normed operator ideals. Our main result provides a complete characterization: a symmetric ideal or symmetric operator space of $τ$-measurable operators satisfies the Fuglede theorem if and only if its commutative core has non-trivial Boyd indices, or equivalently, if it is an interpolation space in the scale of $L_p$-spaces for $1<p<\infty$. This criterion subsumes all previously known cases, including Lorentz and Schatten classes.

math.FA↗

arXiv2025

Uniformity of Maximal Hypoellipticity on Graded Lie Groups: From Pointwise to Global

On graded Lie groups, we develop a mechanism that transfers the uniformity of maximal hypoellipcity from the frozen coefficients principal part of a differential operator to the full operator. Our approach brings the century-old "freeze-unfreeze" strategy into the hypoelliptic setting, and offers a transparent and flexible framework for lifting symbol-level hypoelliptic properties to global elliptic estimates, without relying on pseudodifferential calculus. In addition, we prove that symmetric operators of hypoelliptic type on a graded Lie group are self-adjoint.

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arXiv2025

Singular value asymptotics on compact smooth Riemaniann manifolds

Let $(X,G)$ be a $d$-dimensional compact smooth Riemannian manifold equipped with Laplace-Beltrami operator $Δ_{G}$, and let $Π_{X}$ be the $C^{\ast}$-algebra obtained by locally transferring the $C^{\ast}$-algebra generated by multiplication operators and Riesz transforms on $\mathbb{R}^{d}$. Denote ${\rm sym}_{X}$ the principal symbol mapping of $Π_{X}$. For any $S\inΠ_{X}$, we prove that, in the framework of $C^{\ast}$-algebra, \begin{align*} \lim_{t\rightarrow\infty}t^{\frac{1}{p}}μ(t,S(1+Δ_G)^{-\frac{d}{2p}}) =(2π\sqrt[d]{d})^{-\frac{1}{p}}\Big\|{\rm sym}_{X}(S)\Big\|_{L_{p}(T^{\ast}X,e^{-q_{G}}dλ)}, \end{align*} where $0<p<\infty$, $e^{-q_{G}}$ is the canonical weight on $X$, and $dλ$ is the Liouville measure on the cotangent bundle $T^{\ast}X$.

math.FA↗

arXiv2025

Rank metric isometries and determinant-preserving mappings on II$_1$-factors

We fully describe the general form of a linear (or conjugate-linear) rank metric isometry on the Murray--von Neumann algebra associated with a II$_1$-factor. As an application, we establish Frobenius' theorem in the setting of II$_1$-factors, by showing that every determinant-preserving linear bijection between two II$_1$-factors is necessarily an isomorphism or an anti-isomorphism. This confirms the Harris--Kadison conjecture (1996).

math.OA↗

arXiv2025

Noncommutative sharp dual Doob inequalities

Let $(x_k)_{k=1}^n$ be positive elements in the noncommutative Lebesgue space $L_p(\mathcal{M})$, and let $(\mathcal{E}_k)_{k=1}^n$ be a sequence of conditional expectations with respect to an increasing subalgebras $(\mathcal{M}_n)_{k\geq1}$ of the finite von Neumann algebra $\mathcal{M}$. We establish the following sharp noncommutative dual Doob inequalities: \begin{equation*} \Big\| \sum_{k=1}^nx_k\Big\|_{L_p(\mathcal{M})}\leq \frac{1}{p} \Big\| \sum_{k=1}^n\mathcal{E}_k(x_k)\Big\|_{L_p(\mathcal{M})},\quad 0<p\leq 1, \end{equation*} and \begin{equation*} \Big\| \sum_{k=1}^n\mathcal{E}_k(x_k)\Big\|_{L_p(\mathcal{M})}\leq p\Big\| \sum_{k=1}^nx_k\Big\|_{L_p(\mathcal{M})},\quad 1\leq p\leq 2. \end{equation*} As applications, we obtain several noncommutative martingale inequalities with better constants.

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arXiv2024

Kuroda's theorem for $n$-tuples in semifinite von Neumann algebras

Let $\mathcal{M}$ be a semifinite von Neumann algebra and let $E$ be a symmetric function space on $(0,\infty)$. Denote by $E(\mathcal{M})$ the non-commutative symmetric space of measurable operators affiliated with $\mathcal{M}$ and associated with $E.$ Suppose $n\in \mathbb{N}$ and $E\cap L_{\infty}\not\subset L_{n,1}$, where $L_{n,1}$ is the Lorentz function space with the fundamental function $φ(t)=t^{1/n}$. We prove that for every $\varepsilon>0$ and every commuting self-adjoint $n$-tuple $(α(j))_{j=1}^n,$ where $α(j)$ is affiliated with $\mathcal{M}$ for each $1\leq j\leq n,$ there exists a commuting $n$-tuple $(δ(j))_{j=1}^n$ of diagonal operators affiliated with $\mathcal{M}$ such that $\max\{\|α(j)-δ(j)\|_{E(\mathcal{M})},\|α(j)-δ(j)\|_{\infty}\}<\varepsilon$ for each $1\le j\le n$. In the special case when $\mathcal{M}=B(H)$, our results yield the classical Kuroda and Bercovici-Voiculescu theorems.

math.OA↗

arXiv2024

Spectral asymptotic formula of Bessel--Riesz commutator

Let $R_{λ,j}$ be the $j$-th Bessel--Riesz transform, where $n\geq 1$, $λ>0$, and $j=1,\ldots,n+1$. In this article, we establish a Weyl type asymptotic for $[M_f,R_{λ,j}]$, the commutator of $R_{λ,j}$ with multiplication operator $M_f$, based on building a preliminary result that the endpoint weak Schatten norm of $[M_f,R_{λ,j}]$ can be characterised via homogeneous Sobolev norm $\dot{W}^{1,n+1}(\mathbb{R}_+^{n+1})$ of the symbol $f$. Specifically, the asymptotic coefficient is equivalent to $\|f\|_{\dot{W}^{1,n+1}(\mathbb{R}_+^{n+1})}.$ Our main strategy is to relate Bessel--Riesz commutator to classical Riesz commutator via Schur multipliers, and then to establish the boundedness of Schur multipliers.

math.FA↗

arXiv2024

A $C^{*}$-Algebraic Approach To Principal Symbol Calculus On Filtered Manifolds

From the viewpoint of $*$-homomorphism on $C^{*}$-algebras, we establish the principal symbol mapping for filtered manifolds which are locally isomorphic to stratified Lie groups. Let $\mathbb{G}$ be a stratified Lie group, and let $M$ be a filtered manifold with a $\mathbb{G}$-atlas and a smooth positive density $ν$. For the $C^{*}$-algebra bundle $E_{hom}$ of $M$ constructed from quasi-Riesz transforms on $\mathbb{G}$, we show that there exists a surjective $*$-homomorphism $${\rm sym}_{M}:Π_{M}\to C_{b}(E_{hom})$$ such that $${\rm ker}({\rm sym}_{M})=\mathcal{K}(L_{2}(M,ν))\subset Π_{M}$$ where the domain $Π_{M}\subset\mathcal{B}(L_{2}(M,ν))$ is a $C^{*}$-algebra and $C_{b}(E_{hom})$ is the $C^{*}$-algebra of bounded continuous sections of $E_{hom}$. Especially, we do not make any assumptions on the lattice of the osculating group of $M$ or the assumption of compactness on manifolds in \cite{DAO3,DAO4}.

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