arXiv subjects

Fedor Sukochev

Publications and source records attributed to Fedor Sukochev.

119 records · Page 7Linked to original sources

arXiv2012

(E,F)-multipliers and applications

For two given symmetric sequence spaces $E$ and $F$ we study the $(E,F)$-multiplier space, that is the space all of matrices $M$ for which the Schur product $M\ast A$ maps $E$ into $F$ boundedly whenever $A$ does. We obtain several results asserting continuous embedding of $(E,F)$-multiplier space into the classical $(p,q)$-multiplier space (that is when $E=l_p$, $F=l_q$). Furthermore, we present many examples of symmetric sequence spaces $E$ and $F$ whose projective and injective tensor products are not isomorphic to any subspace of a Banach space with an unconditional basis, extending classical results of S. Kwapień and A. Pełczyński and of G. Bennett for the case when $E=l_p$, $F=l_q$.

math.FA↗

arXiv2012

Measurable operators and the asymptotics of heat kernels and zeta functions

In this note we answer some questions inspired by the introduction, by Alain Connes, of the notion of measurable operators using Dixmier traces. These questions concern the relationship of measurability to the asymptotics of $ζ-$functions and heat kernels. The answers have remained elusive for some 15 years.

math.FA↗

arXiv2011

On the vector-valued Littlewood-Paley-Rubio de Francia inequality

The paper studies Banach spaces satisfying the Littlewood-Paley-Rubio de Francia property LPR_p, 2 \leq p < \infty. The paper shows that every Banach lattice whose 2-concavification is a UMD Banach lattice has this property. The paper also shows that every space having LPR_q also has LPR_p with q \leq p < \infty.

math.FA↗

arXiv2011

Commutator estimates in $W^*$-algebras

Let $\mathcal{M}$ be a $W^*$-algebra and let $LS(\mathcal{M})$ be the algebra of all locally measurable operators affiliated with $\mathcal{M}$. It is shown that for any self-adjoint element $a\in LS(\mathcal{M})$ there exists a self-adjoint element $c_{_{0}}$ from the center of $LS(\mathcal{M})$, such that for any $ε>0$ there exists a unitary element $ u_ε$ from $\mathcal{M}$, satisfying $|[a,u_ε]| \geq (1-ε)|a-c_{_{0}}|$. A corollary of this result is that for any derivation $δ$ on $\mathcal{M}$ with the range in a (not necessarily norm-closed) ideal $I\subseteq\mathcal{M}$, the derivation $δ$ is inner, that is $δ(\cdot)=δ_a(\cdot)=[a,\cdot]$, and $a\in I$. Similar results are also obtained for inner derivations on $LS(\mathcal{M})$.

math.OA↗

arXiv2010

Measure Theory in Noncommutative Spaces

The integral in noncommutative geometry (NCG) involves a non-standard trace called a Dixmier trace. The geometric origins of this integral are well known. From a measure-theoretic view, however, the formulation contains several difficulties. We review results concerning the technical features of the integral in NCG and some outstanding problems in this area. The review is aimed for the general user of NCG.

math.OA↗

arXiv2010

Functions of normal operators under perturbations

In \cite{Pe1}, \cite{Pe2}, \cite{AP1}, \cite{AP2}, and \cite{AP3} sharp estimates for $f(A)-f(B)$ were obtained for self-adjoint operators $A$ and $B$ and for various classes of functions $f$ on the real line $\R$. In this paper we extend those results to the case of functions of normal operators. We show that if a function $f$ belongs to the Hölder class $Ł_\a(\R^2)$, $0<\a<1$, of functions of two variables, and $N_1$ and $N_2$ are normal operators, then $\|f(N_1)-f(N_2)\|\le\const\|f\|_{Ł_\a}\|N_1-N_2\|^\a$. We obtain a more general result for functions in the space $Ł_ø(\R^2)=\big\{f:~|f(\z_1)-f(\z_2)|\le\constø(|\z_1-\z_2|)\big\}$ for an arbitrary modulus of continuity $ø$. We prove that if $f$ belongs to the Besov class $B_{\be1}^1(\R^2)$, then it is operator Lipschitz, i.e., $\|f(N_1)-f(N_2)\|\le\const\|f\|_{B_{\be1}^1}\|N_1-N_2\|$. We also study properties of $f(N_1)-f(N_2)$ in the case when $f\inŁ_\a(\R^2)$ and $N_1-N_2$ belongs to the Schatten-von Neuman class $\bS_p$.

math.FA↗

arXiv2010

Measures from Dixmier Traces and Zeta Functions

For essentially bounded functions on a (closed) compact Riemannian manifold, the noncommutative residue and the Dixmier trace formulation of the noncommutative integral are shown to equate to a multiple of the Lebesgue integral. The identifications are shown to continue to, and be sharp at, square integrable functions. To do better than square integrable, symmetrised noncommutative residue and Dixmier trace formulas are introduced, for which the identifications are shown to continue to $L^{1+ε}$-spaces, $ε> 0$. However, a failure is shown for the Dixmier trace formulation at integrable functions. The (symmetrised) noncommutative residue and Dixmier trace formulas diverge at this point. It is shown the noncommutative residue remains finite and recovers the Lebesgue integral for any integrable function while the Dixmier trace expression can diverge. The results show the claim (in the monograph "Elements of Noncommutative Geometry", Birkhauser, 2001), that the identification on smooth functions obtained using Connes' Trace Theorem can be extended to any integrable function, is false.

math.FA↗

arXiv2010

Functions of perturbed normal operators

In \cite{Pe1}, \cite{Pe2}, \cite{AP1}, \cite{AP2}, and \cite{AP3} sharp estimates for $f(A)-f(B)$ were obtained for self-adjoint operators $A$ and $B$ and for various classes of functions $f$ on the real line $\R$. In this note we extend those results to the case of functions of normal operators. We show that if $f$ belongs to the Hölder class $Ł_\a(\R^2)$, $0<\a<1$, of functions of two variables, and $N_1$ and $N_2$ are normal operators, then $\|f(N_1)-f(N_2)\|\le\const\|f\|_{Ł_\a}\|N_1-N_2\|^\a$. We obtain a more general result for functions in the space $Ł_ø(\R^2)=\big\{f: |f(\z_1)-f(\z_2)|\le\constø(|\z_1-\z_2|)\big\}$ for an arbitrary modulus of continuity $ø$. We prove that if $f$ belongs to the Besov class $B_{\be1}^1(\R^2)$, then it is operator Lipschitz, i.e., $\|f(N_1)-f(N_2)\|\le\const\|f\|_{B_{\be1}^1}\|N_1-N_2\|$. We also study properties of $f(N_1)-f(N_2)$ in the case when $f\inŁ_\a(\R^2)$ and $N_1-N_2$ belongs to the Schatten-von Neuman class $\bS_p$.

math.FA↗

arXiv2010

On a Question of A. E. Nussbaum on Measurability of Families of Closed Linear Operators in a Hilbert Space

The purpose of this note is to answer a question A. E. Nussbaum formulated in 1964 about the possible equivalence between weak measurability of a family of densely defined, closed operators T(t), t real, in a separable complex Hilbert space H on one hand, and the notion of measurability of the 2 \times 2 operator-valued matrix of projections onto the graph Gamma(T(t)) of T(t) on the other, in the negative. Our results demonstrate an interesting distinction between the direct integral over the family of operators T(t) with respect to Lebesgue measure and the naturally maximally defined operator associated with pointwise application of T(t) in the vector-valued Hilbert space L^2(\mathbb R; dt; H). We also provide explicit criteria for the measurability of the matrix of projections onto the graph Gamma(T(t)) of T(t).

math.FA↗

arXiv2009

Operator-Lipschitz functions in Schatten-von Neumann classes

This paper resolves a number of conjectures in the perturbation theory of linear operators. Namely, we prove that every Lipschitz function is operator Lipschitz in the Schatten-von Neumann ideals $S^α$, $1 < α< \infty$. The negative result for $S^α$, $α= 1, \infty$ was earlier established by Yu. Farforovskaya in 1972.

math.FA↗

arXiv2008

Rademacher averages on noncommutative symmetric spaces

Let E be a separable (or the dual of a separable) symmetric function space, let M be a semifinite von Neumann algebra and let E(M) be the associated noncommutative function space. Let $(ε_k)_k$ be a Rademacher sequence, on some probability space $Ω$. For finite sequences $(x_k)_k of E(M), we consider the Rademacher averages $\sum_k ε_k\otimes x_k$ as elements of the noncommutative function space $E(L^\infty(Ω)\otimes M)$ and study estimates for their norms $\Vert \sum_k ε_k \otimes x_k\Vert_E$ calculated in that space. We establish general Khintchine type inequalities in this context. Then we show that if E is 2-concave, the latter norm is equivalent to the infimum of $\Vert (\sum y_k^*y_k)^{1/2}\Vert + \Vert (\sum z_k z_k^*)^{1/2}\Vert$ over all $y_k,z_k$ in E(M) such that $x_k=y_k+z_k$ for any k. Dual estimates are given when E is 2-convex and has a non trivial upper Boyd index. We also study Rademacher averages for doubly indexed families of E(M).

math.OA↗