Suppose $\Cal J$ is a two-sided quasi-Banach ideal of compact operators on a separable infinite-dimensional Hilbert space $\Cal H$. We show that an operator $T\in\Cal J$ can be expressed as finite linear combination of commutators $[A,B]$ where $A\in\Cal J$ and $B\in\Cal B(\Cal H)$ if and only its eigenvalues $(λ_n)$ (arranged in decreasing order of absolute value, repeated according to algebraic multiplicity and augmented by zeros if necessary) satisfy the condition that the diagonal operator $\diag\{\frac1n(λ_1+\cdots +λ_n)\}$ is a member of $\Cal J.$ This answers (for quasi-Banach ideals) a question raised by Dykema, Figiel, Weiss and Wodzicki.
For countably generated ideals, $\Jc$, of $B(\Hil)$, geometric stability is necessary for the canonical spectral characterization of sums of $(\Jc,B(\Hil))$--commutators to hold. This answers a question raised by Dykema, Figiel, Weiss and Wodzicki. There are some ideals, $\Jc$, having quasi--nilpotent elements that are not sums of $(\Jc,B(\Hil))$--commutators. Also, every trace on every geometrically stable ideal is a spectral trace.
We show that elements of unital $C^*$-algebras without tracial states are finite sums of commutators. Moreover, the number of commutators involved is bounded, depending only on the given $C^*$-algebra.
We propose a new class of traces motivated by a trace/trace class property discovered by Laurie, Nordgren, Radjavi and Rosenthal concerning products of operators outside the trace class. Spectral traces, traces that depend only on the spectrum and algebraic multiplicities, possess this property and we suspect others do, but we know of no other traces that do. This paper is intended to be part survey. We provide here a brief overview of some facts concerning traces on ideals, especially involving Lidskii formulas and spectral traces. We pose the central question: whenever the relevant products,…
We study bounded actions of groups and semigroups $G$ on exact sequences of Banach spaces from the point of view of quasilinear maps, characterize the actions on the twisted sum space by commutator estimates and introduce the associated notions of $G$-centralizer and $G$-equivariant map. We will show that when (A) $G$ is an amenable group and (U) the target space is complemented in its bidual by a $G$-equivariant projection, then uniformly bounded compatible families of operators generate bounded actions on the twisted sum space; that compatible quasilinear maps are linear perturbations of $G$…
In this paper, we establish a good-$\lz$ inequality with two parameters in the Schrödinger settings. As it's applications, we obtain weighted estimates for spectral multipliers and Littlewood-Paley operators and their commutators in the Schrödinger settings.
Let $L$ be a non-negative self-adjoint operator on $L^2(X)$. Assume that operator $L$ generates the analytic semigroup $\{e^{-tL}\}_{t>0}$ whose kernels satisfy the standard Gaussian upper bounds. This paper investigates the sharp weighted inequalities for spectral multipliers $F(L)$ and their commutators with BMO functions. The obtained results in this paper can be considered to be improvements of those in \cite{DSY}[Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers, {\it J. Funct. Anal.}, {\bf 260} (2011), 1106-1131].
Let $r_A(T)$ denote the $A$-spectral radius of an operator $T$ which is bounded with respect to the seminorm induced by a positive operator $A$ on a complex Hilbert space $\mathcal{H}$. In this paper, we aim to establish some $A$-spectral radius inequalities for products, sums and commutators of $A$-bounded operators. Moreover, under suitable conditions on $T$ and $A$ we show that \begin{equation*} r_A\left( \sum_{k=0}^{+\infty}c_{k}T^{k}\right) \leq \sum_{k=0}^{+\infty}|c_{k}|\left[r_A(T)\right]^{k}, \end{equation*} where $c_k$ are complex numbers for all $k\in \mathbb{N}$.
In this work we introduce a new measure for the dispersion of the spectral scale of a Hermitian (self-adjoint) operator acting on a separable infinite dimensional Hilbert space that we call spectral spread. Then, we obtain some submajorization inequalities involving the spectral spread of self-adjoint operators, that are related to Tao's inequalities for anti-diagonal blocks of positive operators, Kittaneh's commutator inequalities for positive operators and also related to the Arithmetic-Geometric mean inequality. In turn, these submajorization relations imply inequalities for unitarily invar…
We show a Kalton-Weis type theorem for the general case of non-commuting operators. More precisely, we consider sums of two possibly non-commuting linear operators defined in a Banach space such that one of the operators admits a bounded $H^\infty$-calculus, the resolvent of the other one satisfies some weaker boundedness condition and the commutator of their resolvents has certain decay behavior with respect to the spectral parameters. Under this consideration, we show that the sum is closed and that after a sufficiently large positive shift it becomes invertible, and moreover sectorial. As a…
The paper deals with spectral order isomorphisms between certain spectral sublattices of direct sums of AW*-factors. We prove that these maps consist of spectral order isomorphisms between spectral sublattices of individual direct summands. Consequently, we obtain a complete description of spectral order isomorphisms in the case of atomic AW*-algebras. This includes the setting of matrix algebras. Moreover, we also exhibit the general form of spectral order orthoisomorphisms between various spectral sublattices of direct sums of AW*-factors.
Let $L$ be a non-negative self-adjoint operator acting on the space $L^2(X)$, where $X$ is a metric measure space. Let ${ L}=\int_0^{\infty} λdE_{ L}(λ)$ be the spectral resolution of ${ L}$ and $S_R({ L})f=\int_0^R dE_{ L}(λ) f$ denote the spherical partial sums in terms of the resolution of ${ L}$. In this article we give a sufficient condition on $L$ such that $$ \lim_{R\rightarrow \infty} S_R({ L})f(x) =f(x),\ \ {\rm a.e.} $$ for any $f$ such that ${\rm log } (2+L) f\in L^2(X)$. These results are applicable to large classes of operators including Dirichlet operators on smooth bounded domai…
Let $\G$ be a Carnot group of homogeneous dimension $M$ and $Δ$ its horizontal sublaplacian. For $α\in(0,M)$ we show that operators of the form $H_α:=(-Δ)^α+V$ have no singular spectrum, under generous assumptions on the multiplication operator $V$. The proof is based on commutator methods and Hardy inequalities.
In this article we study localisation of spectral sums $\{S_R\}_{R > 0}$ associated to the sub-Laplacian $\mathcal{L}$ on the Heisenberg Group $\mathbb{H}^d$ where $S_R f := \int_0^R dE_{λ}f$, with $\mathcal{L} = \int_0^{\infty} λ\, dE_λ$ being the spectral resolution of $\mathcal{L}.$ We prove that for any compactly supported function $f \in L^2(\mathbb{H}^d)$, and for any $γ< \frac{1}{2}$, $R^γ S_R f \to 0$ as $ R \to \infty$, almost everywhere off $supp (f)$.
We introduce and study a class of operator tuples in complex Hilbert spaces, which we call spherical tuples. In particular, we characterize spherical multi-shifts, and more generally, multiplication tuples on RKHS. We further use these characterizations to describe various spectral parts including the Taylor spectrum. We also find a criterion for the Schatten $S_p$-class membership of cross-commutators of spherical $m$-shifts. We show, in particular, that cross-commutators of non-compact spherical $m$-shifts cannot belong to $S_p$ for $p \le m$. We specialize our results to some well-studied c…
We study the spectral metric aspects of the standard Podles sphere, which is a homogeneous space for quantum SU(2). The point of departure is the real equivariant spectral triple investigated by Dabrowski and Sitarz. The Dirac operator of this spectral triple interprets the standard Podles sphere as a 0-dimensional space and is therefore not isospectral to the Dirac operator on the 2-sphere. We show that the seminorm coming from commutators with this Dirac operator provides the Podles sphere with the structure of a compact quantum metric space in the sense of Rieffel.
An odd Fredholm module for a given invertible operator on a Hilbert space is specified by an unbounded so-called Dirac operator with compact resolvent and bounded commutator with the given invertible. Associated to this is an index pairing in terms of a Fredholm operator with Noether index. Here it is shown by a spectral flow argument how this index can be calculated as the signature of a finite dimensional matrix called the spectral localizer.
In this short article, we mainly prove that, for any spectral operator $A$ of type $m$ on a complex Hilbert space, if a bounded operator $B$ lies in the collection of bounded linear operators that are in the $k$-centralizer of every bounded linear operator in the $l$-centralizer of $A$, where $k\leqslant l$ is two arbitrary positive integers satisfying $l\geqslant k$ as well as $l\geqslant 2m+1$, then $B$ must belong to the von Neumann algebra generated by $A$ and identity operator. This result generalizes a matrix commutator theorem proved by M.\ F.\ Smiley. For this aim, Smiley-type operator…
We consider the spectral behavior and noncommutative geometry of commutators $[P,f]$, where $P$ is an operator of order $0$ with geometric origin and $f$ a multiplication operator by a function. When $f$ is Hölder continuous, the spectral asymptotics is governed by singularities. We study precise spectral asymptotics through the computation of Dixmier traces; such computations have only been considered in less singular settings. Even though a Weyl law fails for these operators, and no pseudo-differential calculus is available, variations of Connes' residue trace theorem and related integral fo…
