Szego limit theorem for operators with discontinuous symbols and applications to entanglement entropy

Szegö type limit theorems on the Heisenberg group

Let $\mathcal{H}=-Δ_{\mathbb{H}}+V$ be the Schrödinger operator on the Heisenberg group $\mathbb{H}^n$, where $Δ_{\mathbb{H}}$ is the full laplacian on $\mathbb{H}^n$ and $V$ is a positive smooth potential, bounded below and grows like $|g|^κ, κ>0$ for large $|g|$. Let $\mathcal{P}_{r}$ be the orthogonal projection of $L^2(\mathbb{H}^n)$ onto the space of eigenfunctions of $\mathcal{H}$ with eigenvalue $\leq r$; Let $A$ be a 0-th order self-adjoint pseudo-differential operator on $L^2(\mathbb{H}^n)$ relative to the operator $1+|λ|H+V(g), g\in \mathbb{H}^n, λ\in \mathbb{R}^*$ with symbol $a(g, …

Functional Models and Minimal Contractive Liftings

Based on a careful analysis of functional models for contractive multi-analytic operators we establish a one-to-one correspondence between unitary equivalence classes of minimal contractive liftings of a row contraction and injective symbols of contractive multi-analytic operators. This allows an effective construction and classification of all such liftings with given defects. Popescu's theory of characteristic functions of completely non-coisometric row contractions is obtained as a special case satisfying a Szegö condition. In another special case of single contractions and defects equal to…

Bergman和Fock空间上的定位算子

Localization operators on Bergman and Fock spaces

本文研究了Bergman和Fock空间上的定位算子，证明其在特定条件下的弱收敛性，并推导了相关应用，包括Toeplitz算子的范数估计、加窗Berezin变换和Szegö型定理。

Functions of self-adjoint operators in ideals of compact operators

For self-adjoint operators $A, B$, a bounded operator $J$, and a function $f:\mathbb R\to\mathbb C$ we obtain bounds in quasi-normed ideals of compact operators for the difference $f(A)J-Jf(B)$ in terms of the operator $AJ-JB$. The focus is on functions $f$ that are smooth everywhere except for finitely many points. A typical example is the function $f(t) = |t|^γ$ with $γ\in (0, 1)$. The obtained results are applied to derive a two-term quasi-classical asymptotic formula for the trace $tr f(S)$ with $S$ being a Wiener-Hopf operator with a discontinuous symbol.

Essential Spectra of Quasi-parabolic Composition Operators on Hardy Spaces of Analytic Functions

In this work we study the essential spectra of composition operators on Hardy spaces of analytic functions which might be termed as "quasi-parabolic". This is the class of composition operators on H^{2} with symbols whose conjugate with the Cayley transform on the upper half-plane are of the form ϕ(z) = z+ψ(z) where ψ\in H^{2}(\mathbb{H}) and \Im(ψ(z)) >δ> 0. We especially examine the case where ψis discontinuous at infinity. A new method is devised to show that this type of composition operators fall in a C*-algebra of Toeplitz operators and Fourier multipliers. This method enables us to prov…

伪微分算子与Orlicz调制空间及熵估计

Pseudo-differential calculi and entropy estimates with Orlicz modulation spaces

推导了符号属于Orlicz调制空间的伪微分算子在这些空间上的连续性性质，推广了已知结果，并证明了熵泛函在介于$M^p$与$M^2$（$p<2$）的合适Orlicz调制空间上的连续性。

Quasi-parabolic Composition Operators on Weighted Bergman Spaces

In this work we study the essential spectra of composition operators on weighted Bergman spaces of analytic functions which might be termed as "quasi-parabolic." This is the class of composition operators on $A_α^{2}$ with symbols whose conjugate with the Cayley transform on the upper half-plane are of the form $φ(z)=$ $z+ψ(z)$, where $ψ\in$ $H^{\infty}(\mathbb{H})$ and $\Im(ψ(z)) > ε> 0$. We especially examine the case where $ψ$ is discontinuous at infinity. A new method is devised to show that this type of composition operators fall in a C*-algebra of Toeplitz operators and Fourier multiplie…