Schatten-von Neumann Class Research Articles

Asymptotics of Eigenvalues for Differential Operators of Fractional Order

Abstract In this paper we deal with a linear combination of a second order uniformly elliptic operator and the Kipriyanov fractional differential operator. We use a novel method based on properties of a real component to study such type of operators. We conduct the classification of the operators by belonging of their resolvent to the Schatten-von Neumann class and formulate the sufficient condition for the completeness of the root functions system. Finally we obtain an asymptotic formula.

The windowed Fourier transform and Gabor multipliers associated with the Riemann-Liouvlle transform

ABSTRACTWe define the windowed Fourier transform, called also the Gabor transform, associated with singular partial differential operators. We establish Plancherel theorem, orthogonality property and inversion formula for this transform. Next, we define the localization operators , called also Gabor multipliers, associated with two windows and with symbol σ. First, we study the boundedness and compactness of the operators . Last, we define the Schatten-von Neumann class and we prove that the localization operators belong to .

On the complete bounds of L_p-Schur multipliers

We study the class mathcal {M}_p of Schur multipliers on the Schatten-von Neumann class mathcal {S}_p with 1 le p le infty as well as the class of completely bounded Schur multipliers mathcal {M}^{cb}_p. We first show that for 2 le p < q le infty there exists m in mathcal {M}^{cb}_p with m not in mathcal {M}_q, so in particular the following inclusions that follow from interpolation are strict: mathcal {M}_q subsetneq mathcal {M}_p and mathcal {M}^{cb}_q subsetneq mathcal {M}^{cb}_p. In the remainder of the paper we collect computational evidence that for pnot = 1,2, infty we have mathcal {M}_p = mathcal {M}^{cb}_p, moreover with equality of bounds and complete bounds. This would suggest that a conjecture raised by Pisier (Astérisque 247:vi+131, 1998) is false.

Spectral properties of harmonic Toeplitz operators and applications to the perturbed Krein Laplacian

We consider harmonic Toeplitz operators $T_V = PV:{\mathcal H}(\Omega) \to {\mathcal H}(\Omega)$ where $P: L^2(\Omega) \to {\mathcal H}(\Omega)$ is the orthogonal projection onto ${\mathcal H}(\Omega) = \left\{u \in L^2(\Omega)\,|\,\Delta u = 0 \; \mbox{in}\;\Omega\right\}$, $\Omega \subset {\mathbb R}^d$, $d \geq 2$, is a bounded domain with $\partial \Omega \in C^\infty$, and $V: \Omega \to {\mathbb C}$ is a suitable multiplier. First, we complement the known criteria which guarantee that $T_V$ is in the $p$th Schatten-von Neumann class $S_p$, by sufficient conditions which imply $T_V \in S_{p, {\rm w}}$, the weak counterpart of $S_p$. Next, we assume that $\Omega$ is the unit ball in ${\mathbb R}^d$, and $V = \overline{V}$ is radially symmetric, and investigate the eigenvalue asymptotics of $T_V$ if $V$ has a power-like decay at $\partial \Omega$ or $V$ is compactly supported in $\Omega$. Further, we consider general $\Omega$ and $V \geq 0$ which is regular in $\Omega$, and admits a power-like decay of rate $\gamma > 0$ at $\partial \Omega$, and we show that in this case $T_V$ is unitarily equivalent to a pseudo-differential operator of order $-\gamma$, self-adjoint in $L^2(\partial \Omega)$. Using this unitary equivalence, we obtain the main asymptotic term of the eigenvalue counting function for the operator $T_V$. Finally, we introduce the Krein Laplacian $K \geq 0$, self-adjoint in $L^2(\Omega)$; it is known that ${\rm Ker}\,K = {\mathcal H}(\Omega)$, and the zero eigenvalue of $K$ is isolated. We perturb $K$ by $V \in C(\overline{\Omega};{\mathbb R})$, and show that $\sigma_{\rm ess}(K+V) = V(\partial \Omega)$. Assuming that $V \geq 0$ and $V{|\partial \Omega} = 0$, we study the asymptotic distribution of the eigenvalues of $K \pm V$ near the origin, and find that the effective Hamiltonian which governs this distribution is the Toeplitz operator $T_V$.

A Schatten–von Neumann class criterion for the magnetic Weyl calculus

We prove a criterion for a “magnetic” Weyl operator to be trace-class by extending a method developed by Cordes, Kato and Arsu. Using the Calderon–Vaillancourt type Theorem for magnetic Weyl operators and an interpolation argument we also give a criterion for a “magnetic” Weyl operator to be in a Schatten–von Neumann class with .

Toeplitz Operators for Wavelet Transform Related to the Spherical Mean Operator

We define wavelets and wavelet transforms associated with spherical mean operator. We establish a Plancherel theorem, orthogonality property and inversion formula for the wavelet transform. Next, we define the Toeplitz operators \(\mathfrak {T}_{\varphi ,\psi }(\sigma )\) associated with two wavelets \(\varphi ,\psi \) and with symbol \(\sigma .\) We establish the boundedness and compactness of these operators. Last, we define the Schatten-von Neumann class \(S^p\ ;\ p\in \ [1,+\infty ],\) and we show that the Toeplitz operators belong to the class \(S^p\) and we prove a formula of trace.

Conditions of Spectrum Localization for Operators not Close to Self-Adjoint Operators

The Keldysh theorem is generalized to an arbitrary closed operator that is not necessarily close to self-adjoint operators and has a resolvent of Schatten–von Neumann class S p . Based on this theorem, conditions of spectrum localization are obtained for certain classes of non-self-adjoint differential operators.

Time–Frequency Concentration, Heisenberg Type Uncertainty Principles and Localization Operators for the Continuous Dunkl Wavelet Transform on $$\mathbb {R}^{d}$$ R d

We consider the continuous Dunkl wavelet transform $$\Phi ^{D}_{h} $$ associated with the Dunkl operators on $$\mathbb {R}^{d}$$ . We analyze the concentration of this transform on sets of finite measure. In particular, Donoho–Stark and Benedicks type uncertainty principles are given. Next, we prove many versions of Heisenberg type uncertainty principles for $$\Phi ^{D}_{h} $$ . Quantitative Shapiro’s dispersion uncertainty principle and umbrella theorem are proved for the Dunkl continuous wavelet transform. Finally, we investigate the localization operators for $$\Phi ^{D}_{h} $$ , in particular we prove that they are in the Schatten-von Neumann class.

The Donoho–Stark, Benedicks and Heisenberg type uncertainty principles, and the localization operators for the Heckman–Opdam continuous wavelet transform on $${\mathbb {R}}^{d}$$ R d

We consider the continuous wavelet transform \(\Phi _{h}^{W}\) associated with the Heckman–Opdam operators on \(\mathbb {R}^{d}\). We analyse the concentration of this transform on sets of finite measure. In particular, Donoho–Stark and Benedicks-type uncertainty principles are given. Next, we prove many versions of Heisenberg-type uncertainty principles for \(\Phi _{h}^{W}\). Finally, we investigate the localization operators for \(\Phi _{h}^{W}\), in particular we prove that they are in the Schatten–von Neumann class.

Q-Frequent hypercyclicity in spaces of operators

We provide conditions for a linear map of the form $$C_{R,T}(S)=RST$$ to be q-frequently hypercyclic on algebras of operators on separable Banach spaces. In particular, if R is a bounded operator satisfying the q-frequent hypercyclicity criterion, then the map $$C_{R}(S)=RSR^*$$ is shown to be q-frequently hypercyclic on the space $$\mathcal {K}(H)$$ of all compact operators and the real topological vector space $$\mathcal {S}(H)$$ of all self-adjoint operators on a separable Hilbert space H. Further we provide a condition for $$C_{R,T}$$ to be q-frequently hypercyclic on the Schatten von Neumann classes $$S_p(H)$$ . We also characterize frequent hypercyclicity of $$C_{M^*_\varphi ,M_\psi }$$ on the trace-class of the Hardy space, where the symbol $$M_\varphi $$ denotes the multiplication operator associated to $$\varphi $$ .

Localization operators of the wavelet transform associated to the Riemann–Liouville operator

We study the continuous wavelet transform [Formula: see text] associated with the Riemann–Liouville operator. Next, we investigate the localization operators for [Formula: see text]; in particular we prove that they are in the Schatten-von Neumann class.

Time-frequency analysis of localization operators associated to the windowed Hankel transform

ABSTRACTIn the present paper, we define and study a class of pseudodifferential operators known as time-frequency analysis of localization operators which depend on a symbol σ and two windows functions and . We give criteria in terms of a symbol σ for its boundness. We also show that these operators are in Schatten–von Neumann class. Besides, we give a trace formula.

On the Schatten–von Neumann properties of some pseudo-differential operators

We obtain a number of explicit estimates for quasi-norms of pseudo-differential operators in the Schatten–von Neumann classes Sq with 0<q⩽1. The estimates are applied to derive semi-classical bounds for operators with smooth or non-smooth symbols.

Special operator classes and their properties

We introduce some special operator classes and study in terms of Berezin symbols their properties. In particular, we give some characterizations of compact operators and Schatten-von Neumann class operators in terms of Berezin symbols. We also consider some classes of compact operators on a Hilbert space $H,$ which are generalizations of the well known Schatten-von Neumann classes of compact operators. Namely, for any number $p \in (0,\infty)$ and the sequence $w:=(w_{n})_{n\geq0}$ of complex numbers $w_{n},$ $n\geq 0,$ we define the following classes of compact operators on $H $: $$S_{p}^{w}(H)=\left\{ K\in S_{\infty}(H):\sum_{n=0}^{\infty}(s_{n} (K))^{p}w_{n}^{p}\hbox{ is convergent series }\right\}, $$ where $s_{n}(K)$ denotes the $n$th singular number of the operator $K$. The characterizations of these classes are given in terms of Berezin symbols.

Compactness and Berezin symbols

We answer a question raised by Nordgren and Rosenthal about the Schatten-von Neumann class membership of operators in standard reproducing kernel Hilbert spaces in terms of their Berezin symbols.

Functions of perturbed tuples of self-adjoint operators

We generalize earlier results of Aleksandrov and Peller (2010) [2,3], Aleksandrov et al. (2011) [6], Peller (1985) [13], Peller (1990) [14] to the case of functions of n-tuples of commuting self-adjoint operators. In particular, we prove that if a function f belongs to the Besov space B∞,11(Rn), then f is operator Lipschitz and we show that if f satisfies a Hölder condition of order α, then ‖f(A1,…,An)−f(B1,…,Bn)‖⩽constmax1⩽j⩽n‖Aj−Bj‖α for all n-tuples of commuting self-adjoint operators (A1,…,An) and (B1,…,Bn). We also consider the case of arbitrary moduli of continuity and the case when the operators Aj−Bj belong to the Schatten–von Neumann class Sp.

Functions of perturbed dissipative operators

We generalize our results of \cite{AP2} and \cite{AP3} to the case of maximal dissipative operators. We obtain sharp conditions on a function analytic in the upper half-plane to be operator Lipschitz. We also show that a Holder function of order $\a$, $0<\a<1$, that is analytic in the upper half-plane must be operator Holder of order $\a$. Then we generalize these results to higher order operator differences. We obtain sharp conditions for the existence of operator derivatives and express operator derivatives in terms of multiple operator integrals with respect to semi-spectral measures. Finally, we obtain sharp estimates in the case of perturbations of Schatten-von Neumann class $\bS_p$ and obtain analogs of all the results for commutators and quasicommutators. Note that the proofs in the case of dissipative operators are considerably more complicated than the proofs of the corresponding results for self-adjoint operators, unitary operators, and contractions that were obtained earlier in \cite{AP2}, \cite{AP3}, and \cite{Pe6}.

On trace and Hilbert-Schmidt norm estimates

Let ℰ and 풫 be nonnegative quadratic forms in the Hilbert space ℋ. Suppose that, for every β ⩾ 0, the form ℰ+β풫 is densely defined and closed. Let Hβ be the self-adjoint operator associated with ℰ+β 풫 and R∞ := lim β→∞ (Hβ+1)−1. We give estimates for the distance between (Hβ+1)−1 and R∞ with respect to the norm ‖·‖p in the Schatten–von Neumann class of order p, p=1, 2. In particular, we derive a condition that is necessary and sufficient in order that ‖ ( H β + 1 ) − 1 − R ∞ ‖ 1 ⩽ c / β ∀ β > 0 for some finite constant c, and give examples where this criterion is satisfied.

Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group

Abstract We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue– Orlicz spaces of a discrete group and relative Toeplitz-Schur multipliers on Schatten–von- Neumann–Orlicz classes. Four applications are given: lacunary sets, unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum , the norm of the Hilbert transformand the Riesz projection on Schatten–von-Neumann classes with exponent a power of 2, and the norm of Toeplitz Schur multipliers on Schatten–von-Neumann classes with exponent less than 1.

On perturbations of the isometric semigroup of shifts on the semiaxis

We study perturbations $(\tilde\tau_t)_{t\ge 0}$ of the semigroup of shifts $(\tau_t)_{t\ge 0}$ on $L^2(\R_+)$ with the property that $\tilde\tau_t - \tau_t$ belongs to a certain Schatten-von Neumann class $\gS_p$ with $p\ge 1$. We show that, for the unitary component in the Wold-Kolmogorov decomposition of the cogenerator of the semigroup $(\tilde\tau_t)_{t\ge 0}$, {\it any singular} spectral type may be achieved by $\gS_1$ perturbations. We provide an explicit construction for a perturbation with a given spectral type based on the theory of model spaces of the Hardy space $H^2$. Also we show that we may obtain {\it any} prescribed spectral type for the unitary component of the perturbed semigroup by a perturbation from the class $\gS_p$ with $p>1$.