Searcharxiv⌕ Search

arXiv subjects

Yufeng Lu

Publications and source records attributed to Yufeng Lu.

At least 37 records · Page 2Linked to original sources
arXiv2026

Spectral properties of Toeplitz operators with harmonic function symbols on the Bergman space

This paper investigates the spectral properties of Toeplitz operators on the Bergman space of unit disk. We present an integral representation of $ T^*_{z^m}$, which establishes a connection between the Bergman functions and the solutions of PDE theory. In fact, by leveraging the Poincaré theorem in difference equations and the solution forms of differential equations, this paper describes the kernels of certain Toeplitz operators with harmonic polynomial symbols, and further gives the sufficient conditions for the connectedness of the spectra of these Toeplitz operators. The spectral properties of $ T_φ$ with $φ(z) =\overline{z}^{m} + αz^m + β$ are characterized, such as $σ(T_φ)= \overline{φ(\mathbb {D})}$, Fredholm index of $T_φ$ can only be one of $m,-m$ and $0$, $T_φ$ satisfies Coburn's theorem. These findings offer an illuminating example for the essential projective spectra of non-commuting operators.

math.FA↗
arXiv2026

Spectral dynamics for the infinite dihedral group and the lamplighter group

For a tuple $A=(A_0,A_1,\cdots,A_n)$ of elements in a Banach algebra $\mathfrak{B}$, its projective (joint) spectrum $p(A)$ is the collection of $z\in \mathbb{P}^n$ such that $A(z)=z_0A_0+z_1A_1+\cdots+z_nA_n$ is not invertible. If $\mathfrak{B}$ is the group $C^*$-algebra for a discrete group $G$ generated by $A_0, A_1,\dots, A_n$ with a representation $ρ$, then $p(A)$ is an invariant of (weak) equivalence for $ρ$. In \cite{BY}, B. Goldberg and R. Yang proved that the Julia set $\mathcal{J}(F)$ of the induced rational map $F$ for the infinite dihedral group $D_\infty$ is the union of the projective spectrum with the extended indeterminacy set. But the extended indeterminacy set $E_F$ is complicated. To obtain a better relationship between the projective spectrum and the Julia set, by replacing $A_π(z)=z_0+z_1π(a)+z_2π(t)$ with the extended pencil $A_π(z)=z_0+z_1π(a)+z_2π(t)+z_3π(at)$, where $π$ is the Koopman representation, and using the method of operator recursions, we show that $p(A_π)=\mathcal{J}(F).$ Further, we study the spectral dynamics for the Lamplighter group $\mathcal{L}$, and prove that $\mathcal{J}(Q)=E_Q$, where $Q$ is the rational map associated with $\mathcal{L}$.

math.FA↗
arXiv2026

A geometric approach to the compressed shift operator on the Hardy space over the bidisk

This paper studies the compressed shift operator $S_z$ on the Hardy space over the bidisk via the geometric approach. We calculate the spectrum and essential spectrum of $S_z$ on the Beurling type quotient modules induced by rational inner functions, and give a complete characterization for $S_z^*$ to be a Cowen-Douglas operator. Then we extend the concept of Cowen-Douglas operator to be the generalized Cowen-Douglas operator, and show that $S_z^*$ is a generalized Cowen-Douglas operator. Moreover, we establish the connection between the reducibility of the Hermitian holomorphic vector bundle induced by kernel spaces and the reducibility of the generalized Cowen-Douglas operator. By using the geometric approach, we study the reducing subspaces of $S_z$ on certain polynomial quotient modules.

math.FA↗
arXiv2025

Global well-posedness for generalized fractional Hartree equations with rough initial data in all dimensions

We prove the global existence of the solution for fractional Hartree equations with initial data in certain real interpolation spaces between $L^{2}$ and some kinds of new function spaces defined by fractional Schrödinger semigroup, which could imply the global well-posedness of the equation in modulation spaces $M_{p,p'}^{s_{p}}$ for $p$ close to 2 with no smallness condition on initial data, where $s_{p}=(m-2)(1/2-1/p)$. The proof adapts a splitting method inspired by the work of Hyakuna-Tsutsumi, Chaichenets et al. to the modulation spaces and exploits polynomial growth of the fractional Schrödinger semi-group on modulation spaces $M_{p,p'}$ with loss of regularity $s_{p}$.

math.AP↗
arXiv2025

Essential norm and Schatten class of Hankel operators between weighted Fock spaces

In this paper, we characterize the essential norm of Hankel operators from a weighted Fock space $F_φ^{p}$ to a weighted Lebesgue space $L_φ^{q}$ for all $1 \leq p, q < \infty$. Additionally, we characterize the Schatten-$h$ class membership for Hankel operators acting from $F_φ^{2}$ to $L_φ^{2}$, where the class is defined by a continuous increasing convex function $h$.

math.FA↗
arXiv2025

Notes on a special order on $\mathbb{Z}^\infty$

In 1958, Helson and Lowdenslager extended the theory of analytic functions to a general class of groups with ordered duals. In this context, analytic functions on such a group $G$ are defined as the integrable functions whose Fourier coefficients lie in the positive semigroup of the dual of $G$. In this paper, we found some applications of their theory to infinite-dimensional complex analysis. Specifically, we considered a special order on $\mathbb{Z}^\infty$ and corresponding analytic continuous functions on $\mathbb{T}^ω$, which serves as the counterpart of the disk algebra in infinitely many variables setting. By characterizing its maximal ideals, we have generalized the following theorem to the infinite-dimensional case: For a positive function $w$ that is integrable and log-integrable on $\mathbb{T}^d$, there exists an outer function $g$ such that $w=|g|^2$ if and only if the support of $\hat{\log w}$ is a subset of $\mathbb{N}^d\cap (-\mathbb{N})^d$. Furthermore, we have found the counterpart of the above function algebra in the closed right half-plane, and the representing measures of each point in the right half-plane for this algebra. As an application of the order, we provided a new proof of the infinite-dimensional Szegö's theorem.

math.FA↗
arXiv2025

Dynamic real-time multi-UAV cooperative mission planning method under multiple constraints

As UAV popularity soars, so does the mission planning associated with it. The classical approaches suffer from the triple problems of decoupled of task assignment and path planning, poor real-time performance and limited adaptability. Aiming at these challenges, this paper proposes a dynamic real-time multi-UAV collaborative mission planning algorithm based on Dubins paths under a distributed formation structure. Dubins path with multiple advantages bridges the gap between task assignment and path planning, leading to a coupled solution for mission planning. Then, a series of acceleration techniques, task clustering preprocessing, highly efficient distance cost functions, low-complexity and less iterative task allocation strategies, are employed to guarantee the real-time performance of the algorithms. To cope with different emergencies and their simultaneous extremes, real-time planning of emerging tasks and mission replanning due to the reduction of available UAVs are appropriately handled. Finally, the developed algorithm is comprehensively exemplified and studied through simulations, highlighting that the proposed method only sacrifices 9.57% of the path length, while achieving a speed improvement of 4-5 orders of magnitude over the simulated annealing method, with a single mission planning of about 0.0003s.

cs.RO↗
arXiv2025

Which hyponormal block Toeplitz operators are either normal or analytic?

In this paper, we continue Curto-Hwang-Lee's work to study the connection between hyponormality and subnormality for block Toeplitz operators acting on the vector-valued Hardy space of the unit circle. Curto-Hwang-Lee's work focuses primarily on hyponormality and subnormality of block Toeplitz operators with rational symbols. By studying the greatest common divisor of matrix-valued inner functions and the ``weak" commutativity of matrix-valued inner functions, we extended Curto-Hwang-Lee's result to block Toeplitz operators with symbols of bounded type. More precisely, we proved that if $Ψ,Ψ^{\ast}$ are matrix-valued functions of bounded type and the inner part of $Ψ$ of Douglas-Shapiro-Shields factorization is a scalar inner function, then every hyponormal Toeplitz operator $T_Ψ$ whose square is also hyponormal must be either normal or analytic.

math.FA↗
arXiv2025

Hilbert-Schmidtness of the $M_{θ,φ}$-type submodules

Let $θ(z),φ(w)$ be two nonconstant inner functions and $M$ be a submodule in $H^2(\mathbb{D}^2)$. Let $C_{θ,φ}$ denote the composition operator on $H^2(\mathbb{D}^2)$ defined by $C_{θ,φ}f(z,w)=f(θ(z),φ(w))$, and $M_{θ,φ}$ denote the submodule $[C_{θ,φ}M]$, that is, the smallest submodule containing $C_{θ,φ}M$. Let $K^M_{λ,μ}(z,w)$ and $K^{M_{θ,φ}}_{λ,μ}(z,w)$ be the reproducing kernel of $M$ and $M_{θ,φ}$, respectively. By making full use of the positivity of certain de Branges-Rovnyak kernels, we prove that \[K^{M_{θ,φ}}= K^M \circ B~ \cdot R,\] where $B=(θ,φ)$, $R_{λ,μ}(z,w)=\frac{1-\overline{θ(λ)}θ(z)}{1-\barλz} \frac{1-\overline{φ(μ)}φ(w)}{1-\barμw}$. This implies that $M_{θ,φ}$ is a Hilbert-Schmidt submodule if and only if $M$ is. Moreover, as an application, we prove that the Hilbert-Schmidt norms of submodules $[θ(z)-φ(w)]$ are uniformly bounded.

math.FA↗
arXiv2024

Global Well-posedness for the Fourth-order Nonlinear Schrodinger Equation

The local and global well-posedness for the one dimensional fourth-order nonlinear Schrödinger equation are established in the modulation space $M^{s}_{2,q}$ for $s\geq \frac12$ and $2\leq q <\infty$. The local result is based on the $U^p-V^p$ spaces and crucial bilinear estimates. The key ingredient to obtain the global well-posedness is that we achieve a-priori estimates of the solution in modulation spaces by utilizing the power series expansion of the perturbation determinant introduced by Killip-Visan-Zhang for completely integrable PDEs.

math.AP↗
arXiv2023

The dissipativity and controllability of input affine systems

The paper is considered with the dissipative theory and feedback control under the framework of dissipation with the supply rate is the inner product of input u and the derivation of output y for the linear/nonlinear time-invariant input affine system. A necessary and sufficient condition of dissipativity of the class of systems is presented. Furthermore, we obtain necessary and sufficient conditions of the negative imaginary property, input strictly negative imaginary property and output strictly negative imaginary property for the nonlinear input affine system, which can cover the case that the linear time-invariant system in the literature. Besides, under given conditions, we consider the equivalence between the negative imaginary property and input-output stability. In other words, the system with negative imaginary property may be input-output stable under some certain conditions; conversely, the input-output stable system may also have negative imaginary property. Last but not least, we consider the equivalence between the asymptotical stability and strict dissipativity with the supply rate and some dissipative matrices are given for a certain linear time-invariant system.

math.DS↗
arXiv2022

Geometry of weighted Lorentz-Finsler manifolds II: A splitting theorem

We show an analogue of the Lorentzian splitting theorem for weighted Lorentz-Finsler manifolds: If a weighted Berwald spacetime of nonnegative weighted Ricci curvature satisfies certain completeness and metrizability conditions and includes a timelike straight line, then it necessarily admits a one-dimensional family of isometric translations generated by the gradient vector field of a Busemann function. Moreover, our formulation in terms of the $ε$-range introduced in our previous work enables us to unify the previously known splitting theorems for weighted Lorentzian manifolds by Case and Woolgar-Wylie into a single framework.

math.DG↗
arXiv2022

Sharp embedding between Wiener amalgam and some classical spaces

We establish the sharp conditions for the embedding between Wiener amalgam spaces $W_{p,q}^s$ and some classical spaces, including Sobolev spaces $L^{s,r}$, local Hardy spaces $h_{r}$, Besov spaces $B_{p,q}^s$, which partially improve and extend the main result obtained by Guo et al. in J. Funct. Anal., 273(1):404-443, 2017. In addition, we give the full characterization of inclusion between Wiener amalgam spaces $W_{p,q}^s$ and $α$-modulation spaces $M_{p,q}^{s,α}$. Especially, in the case of $α=0$ with $M_{p,q}^{s,α} = M_{p,q}^s$, we give the sharp conditions of the most general case of these embedding. When $0<p\leqslant 1$, we also establish the sharp embedding between Wiener amalgam spaces and Triebel spaces $F_{p,r}^{s}$.

math.FA↗
arXiv2022

Local smoothing estimates of fractional Schrödinger equations in $α$-modulation spaces with some applications

We show some new local smoothing estimates of the fractional Schrödinger equations with initial data in $α$-modulation spaces via decoupling inequalities. Furthermore, our necessary conditions show that the local smoothing estimates are sharp in some cases. As applications, the local smoothing estimates could show some new local well-posedness on modulation spaces of the fourth-order nonlinear Schrödinger equations on the line.

math.AP↗
arXiv2022

Volume comparison theorems in Finsler spacetimes

In a $(1+n)$-dimensional Lorentz--Finsler manifold with $N$-Bakry--Émery Ricci curvature bounded below for $N\in(n,\infty]$, using the Riccati equation techniques, we established the Bishop--Gromov volume comparison for the so-called standard sets for comparisons in Lorentzian volumes (SCLVs).We also established the Günther volume comparison for SCLVs when the flag curvature was bounded above.

math.DG↗
arXiv2022

Global Cauchy problems for the nonlocal (derivative) NLS in $E^s_σ$

We consider the Cauchy problem for the (derivative) nonlocal NLS in super-critical function spaces $E^s_σ$ for which the norms are defined by $$ \|f\|_{E^s_σ} = \|\langleξ\rangle^σ2^{s|ξ|}\hat{f}(ξ)\|_{L^2}, \ s<0, \ σ\in \mathbb{R}. $$ Any Sobolev space $H^{r}$ is a subspace of $E^s_σ$, i.e., $H^r \subset E^s_σ$ for any $ r,σ\in \mathbb{R}$ and $s<0$. Let $s<0$ and $σ>-1/2$ ($σ>0$) for the nonlocal NLS (for the nonlocal derivative NLS). We show the global existence and uniqueness of the solutions if the initial data belong to $E^s_σ$ and their Fourier transforms are supported in $(0, \infty)$, the smallness conditions on the initial data in $E^s_σ$ are not required for the global solutions.

math.AP↗
arXiv2022

Comparison theorems on weighted Finsler manifolds and spacetimes with $ε$-range

We establish the Bonnet-Myers theorem, Laplacian comparison theorem, and Bishop-Gromov volume comparison theorem for weighted Finsler manifolds as well as weighted Finsler spacetimes, of weighted Ricci curvature bounded below by using the weight function. These comparison theorems are formulated with $ε$-range introduced in our previous paper, that provides a natural viewpoint of interpolating weighted Ricci curvature conditions of different effective dimensions. Some of our results are new even for weighted Riemannian manifolds and generalize comparison theorems of Wylie-Yeroshkin and Kuwae-Li.

math.DG↗