Papers

We obtain the relation between non-hermite skew information $NI_{\rho,\alpha}(A)$, $NU_{\rho,\alpha}(A)$, variance $NV_{\rho}(A)$ and $NJ_{\rho,\alpha}(A)$ by using the maximal eigenvalue and the minimal eigenvalue of density $\rho$. And also we obtain their refinements.

We investigate the existence of ground states for two-component Bose–Einstein condensates with intraspecies interactions $a_1, a_2 \in (0, a^*)$ and interspecies interaction $\beta>0$. By carefully investigating an associated auxiliary minimization problem, we prove the existence of a unique, continuous, critical surface $\gamma = \gamma(a_1, a_2)$ between $\beta_*:= \sqrt{(a^* - a_1) (a^* - a_2)}$ and $\beta^*:= a^* - \frac{a_1 + a_2}{2}$. This surface provides a complete classification for the existence of minimizers: ground states exist for $\beta \in (0,\gamma)$ and do not exist for $\beta > \gamma$. Additionally, when the trapping potentials are continuous at a common zero point, there is no ground state for $\beta = \gamma$. Our results close the gap case $\beta \in [\beta_*, \beta^*]$ when $a_1\ne a_2$ in literature, significantly extending the results in \cite{Bao-Cai-1,guoBlowupSolutionsTwo2017a}.

Consider the compressed shift $T=P_{K_\theta}M_z|_{K_\theta}$ on $K_\theta=H^2\ominus\theta H^2$ associated with the atomic singular inner function $\theta(z)=\exp(-a(\zeta+z)/(\zeta-z))$, $a>0$ and $|\zeta|=1$. Let $k_0=P_{K_\theta}1$ and define real powers $T^s$ using a holomorphic logarithm near the singleton spectrum of $T$. For any temporal multisequence $\mathsf S=(s_k)_{k\in I}\subset[0,\infty)$ with uniformly bounded numbers of samples in unit intervals, we prove that $\{T^{s_k}k_0:k\in I\}$ is a frame if and only if $\mathsf S$ is Kummer-thick: the frequencies $\omega(s_k)=\sqrt{8s_k+4}$, weighted by $1/\omega(s_k)$, have uniformly positive total mass in every sufficiently distant interval of some fixed length. Equivalently, the occupied unit cells fill a fixed positive proportion of every window $[N-C\sqrt N,N+C\sqrt N]$ for some $C>0$ and all sufficiently large $N$. The proof combines a Volterra transmutation into Bessel waves with the Fourier–Bessel Logvinenko–Sereda theorem.

In this paper, we consider the strong operator topology (SOT) on the space $\mathcal{L} (X)$ of continuous linear operators on an infinite-dimensional Fréchet space $X$. The existence of SOT-dense subspaces as well as of SOT-closed infinite-dimensional subspaces inside the family of all non-cyclic operators is established. Other classes of operators, such as non-dense range or non-injective operators, are also studied in this regard. Moreover, we prove for the sequence space $\ell_p$ $(0 < p < \infty)$ that the family of all non-cyclic members of $\mathcal{L}(\ell_p)$ contains an isometric copy of $\ell_p$.

Let $n\ge 3$ and let $K\subset\mathbb R^n$ be a convex body. For a two-dimensional linear subspace $P\subset\mathbb R^n$, let $K|P$ be the orthogonal projection of $K$ onto $P$. We prove that if for every two-dimensional subspace $P$, the planar convex body $K|P$ has $q$-fold rotational symmetry up to translation, then for $q\ge4$, this forces $K$ to be an Euclidean ball. The case $q=3$ is exceptional: non-spherical examples exist.

We characterize boundedness of the Bergman projection on operator-weighted $L^p$ spaces over a separable Hilbert space, for $1<p<\infty$. Under the directional integrability conditions, the criterion is uniform boundedness of averages over Carleson squares in the unit disc. We obtain norm estimates independent of dimension. The proof uses a separated kernel expansion and a layer decomposition of the tent tree. No reverse Hölder assumption is needed. We also give a finer matrix estimate with a dimension-dependent constant. Boundedness is unchanged by compactly supported changes of the weight that preserve the integrability conditions.

In this paper, we establish a Bourgain–Brezis–Mironescu formula for BV functions on arbitrary open sets for a general class of radial mollifiers. Building on this formula, we identify the \(\Gamma\)-limit of the improved nonlocal energies with respect to \(L^1\)-convergence as a constant multiple of the total variation. As a consequence, we obtain a characterization of BV functions in terms of the finiteness of the corresponding nonlocal energies. We also establish local \(L^1\)-compactness on arbitrary open sets and global \(L^1\)-compactness on bounded open sets under a natural uniform integrability assumption. In particular, except for the global compactness result, all these results hold on arbitrary open sets, with no assumptions on boundedness, connectedness, or boundary regularity. Counterexamples show that for global compactness, both the boundedness of the domain and the uniform integrability assumption are essential. The proofs rely primarily on one-dimensional restrictions of BV functions and a concentration argument for radial mollifiers.

We give a detailed implementation of the mass-splitting construction of Nazarov, Volberg and Yuditskii for reflectionless measures with singular continuous components. The finite approximants are allowed to have isolated atoms. An exact local operation replaces each active atom by a short permanent interval and two child atoms, thereby eliminating the dependence of a future splitting scale on an already prescribed band width. Summable estimates for completed generations and fixed cylinders retain positive mass on a null Cantor set and exclude atoms in the limit. Uniform analytic estimates near the permanent intervals establish reflectionlessness and exclude endpoint singularities. A separate Green function estimate for thin pairs of intervals gives the Widom condition and regularity of the limiting compact set. The resulting probability measure has both a nonzero absolutely continuous component and a nonzero singular continuous component.

We study adjoints of $R(T)$-linear, $T$-strongly bounded operators on the conditional space $L^2(T)$ and the interaction between its Hilbert-type geometry and its Riesz-space order. This produces two natural operator moduli: the Riesz–Kantorovich order modulus and the spectral modulus $(S^*S)^{1/2}$. After isolating the $T$-regular operators, we show that they form a Dedekind-complete order ideal in the natural lattice of order-bounded $R(T)$-module homomorphisms. On this operator lattice the order operations remain $T$-strongly bounded and the adjoint is an order automorphism. We then compare the two moduli and explain why they may differ.

Nonlocal models have found great success in recent years. In particular, models that replace spatial derivatives with strongly singular integral operators involving finite interaction radii are used in a wide range of applications, including image processing, continuum mechanics, and many other areas. However, from the perspective of mathematical analysis, difficulties arise when considering the appropriate notion of “boundary” conditions. (The nonlocal analogue of the boundary is often called a “collar” and is typically not a lower-dimensional set.) For instance, merely enforcing homogeneous Dirichlet-type constraints on the collar fails to guarantee a number of essential properties used in the analysis and numerical simulation of these models, as we show in the present work. In the local setting, working in a Sobolev space of functions that are zero on the boundary allows one to integrate by parts with no boundary term, approximate by test functions, extend by zero to the full space, and apply the Hardy and Poincaré inequalities. We prove that the nonlocal analog of each of these fails if one only requires the functions to be zero on the collar, and that one must move to a smaller, more restrictive space to enjoy these properties. In addition, we introduce a new lifting operator, correct an error in a previously published result on a nonlocal Green's theorem, and establish how the properties listed above relate to existing notions of fractional Sobolev spaces.

The problem of differentiating Lipschitz curves in Banach spaces goes back to Tamarkin in the early 1930s and is one of the origins of the Radon–Nikodým property in classical Banach space theory. In this article we characterize those quasi-Banach spaces $X$ for which every Lipschitz curve $F\to\mathbb{R}\to X$ admits a point of differentiability and prove that this property holds if and only if $X$ is isomorphic to a Banach space with the Radon–Nikodým property, thus substantiating a conjecture of Kalton, communicated personally to the first-named author in 2008. Equivalently, every nonlocally convex quasi-Banach space admits a nowhere differentiable Lipschitz curve. Our result is obtained from a quantitative characterization of local convexity in terms of the infinitesimal oscillation of Lipschitz curves. Motivated by this obstruction, we then investigate positive differentiability phenomena in $\ell_p$, $0<p<1$, and identify natural decoupled-coordinate constructions for which Lipschitz regularity nevertheless implies almost-everywhere differentiability. We also contrast this behavior with metric differentiability for the natural metric and the quasi-metric on $\ell_p$.

Let $R$ be the closed linear span of the Rademacher functions in $L^1(0,1)$. We prove that $L^1(0,1)/R$ embeds isomorphically into $L^1(0,1)$, answering affirmatively a question of González and Martínez-Abejón \cite[p.~140]{GM97}. We then use this embedding to show that the weak Calkin algebra of the quotient is uniformly incompressible. \end{abstract

We answer three questions from Boulabiar's survey on truncated Riesz spaces. First, the truncation band projection property coincides with the principal projection property, with no completeness or Archimedean hypothesis needed, and the band projection onto a truncation band is then given explicitly by the truncation itself. Second, a truncation on any Riesz space is unital exactly when its fixed-point set has a supremum. Hence bounded truncations on $KB$-spaces are unital, and a nonunital truncated Banach lattice with bounded truncation contains a closed sublattice copy of $c_0$ and is never an $AL$-space, while on the $AM$ side the smallest truncation unitization norm is always an $M$-norm, though the largest one need not be. Third, a truncation homomorphism whose domain truncation is Archimedean need not be a Riesz homomorphism, as an explicit counterexample built from $c_0$ and $c_0/c_{00}$ shows, but it is one modulo truncation infinitesimals whenever the codomain space is Archimedean.

We construct in ZFC a unital, simple, infinite-dimensional $C^*$-algebra whose nonzero irreducible representations form a single unitary equivalence class, but which admits a faithful representation on a separable Hilbert space. The algebra contains a unital copy of the canonical anticommutation relation (CAR) algebra, and the normalized CAR trace has a unique extension among all states. This extension is tracial, and its GNS representation is separable and faithful, with weak closure the hyperfinite $\mathrm{II}_1$ factor. In contrast, every nonzero irreducible representation acts on a Hilbert space of density $2^{\aleph_0}$. The construction separates the added unitaries into shell terms, controlled by finite CAR relations, and rank-one defect terms. These relations determine every irreducible representation of the generated algebra and every extension of the CAR trace. The algebra has norm density $2^{\aleph_0}$; consequently, the continuum hypothesis (CH) is equivalent over ZFC to the existence of a counterexample to Naimark's problem of norm density $\aleph_1$.

We study the closed spans of finitely many joint orbits of holomorphic multipliers on the Hardy space $H^2(\mathbb D^d)$ of the polydisk. If fewer than $d$ symbols are used, every such span has infinite codimension. The symbols may be unbounded, provided that all orbit vectors belong to the Hardy space. We give two elementary proofs. The first uses common level sets and independent point evaluations; for bounded symbols, these evaluations yield joint adjoint eigenvectors. The second uses finite Taylor sections and the nilpotent structure of truncated convolution, also known as the Jury product. For $r$ generators and $q<d$ symbols, the orbit dimension on a cube of side $n$ is $O(n^q)$, whereas the ambient dimension is $n^d$. In the bidisk we obtain the explicit codimension bound $MN-r(M+N-1)$, which is sharp for a single orbit. The coordinate multipliers attain the parameter threshold. We also explain how these obstructions relate to kernel interpolation and the established representation of dynamical frames by compressed shifts on model spaces.

Let \(\mathbf{E}\to M\) be a finite-rank Riemannian vector bundle over a compact Riemannian manifold with nonempty smooth boundary, equipped with a compatible connection. We develop an intrinsic trace theory for Sobolev sections defined through weak covariant derivatives. For every integer \(m\ge1\) and every \(1\le p<\infty\), we construct continuous trace operators for the covariant derivatives up to order \(m-1\) and characterize the Sobolev space with homogeneous Dirichlet boundary conditions as the kernel of the corresponding trace operator. As an application, we establish the existence of ground state solutions for a class of quasilinear elliptic equations on Riemannian vector bundles with generalized \((p,q)\)-growth. The proposed framework unifies several important geometric operators, including the Bochner \(p\)-Laplacian, the \((p,q)\)-Bochner Laplacian, and prescribed mean curvature-type operators.

For every $n\ge2$, we construct $4n-2$ nondegenerate Gaussian measures on $\mathbb{R}^{6n-3}$ that cannot be simultaneously equipartitioned by two affine hyperplanes. This disproves the Ramos conjecture for two hyperplanes. Combined with known upper bounds, the construction shows that $3\cdot2^{s-1}-2$ is the least dimension guaranteeing a common two-hyperplane equipartition of $2^s-2$ absolutely continuous probability measures, for every $s\ge3$. We characterize the Gaussian equipartition threshold in terms of the least number of positive definite quadratic measurements needed for phase retrieval. Modified complex polynomial multiplication gives $2r-2$ positive definite measurements in every even dimension $r\ge4$. This number is optimal when $r=2^k+2$, $k\ge1$.

We provide a purely geometric necessary and sufficient condition, in terms of a certain Cesàro mean of the volume function, for the existence of the Maz'ya-Shaposhnikova functional limit with respect to Gagliardo seminorms on metric measure spaces (under a mild measure condition). As an application, we establish the Maz'ya-Shaposhnikova convergence on the unbounded Sierpiński gasket. We also show that Ahlfors regularity does not imply the existence of this functional limit by constructing a concrete example.

We consider variational image denoising with spatially dependent Orlicz regularization and introduce an Orlicz–Sobolev approximation of the ROF model based on a scaled $L\log L$-type density. For a general class of uniformly superlinear integrands satisfying a $\Delta_2$-condition, we establish existence and uniqueness of minimizers and derive a Fenchel dual problem with dual attainment and pointwise optimality conditions. We then specialize the framework to a logarithmic density whose correction is activated above a prescribed local gradient scale. For this model, we obtain explicit expressions of the Fenchel dual and optimality conditions as well as a pointwise radial formula for the dual proximal map involving the Lambert $W$-function. As the logarithmic parameter tends to zero, we prove equicoercivity and $\Gamma$-convergence to the ROF functional, together with convergence of the corresponding minimizers. Numerical illustrations indicate that the logarithmic correction can reduce staircasing for diffuse transitions, while retaining behavior comparable to ROF for sharp interfaces and a natural image.

Given three normed spaces $X\_0 , X\_1 , \tilde{X\_1}$, with $\tilde{X\_1}$ a closed subspace of $X\_1$ with same norm, how does $[X\_0 , X\_1 ]\_\theta$ compare to $[X\_0 , X\_1 ]\_\theta\,? This classical problem of interpolation theory has been mostly considered in the case of finite codimensional subspaces. We show here that quotient J norms are a useful tool which allows to tackle several relevant cases.

Liu and Ruan showed that a separable operator space has the completely bounded approximation property if and only if it admits a completely bounded frame. We show that, even for Hilbertian operator spaces, such a frame need not give rise to a completely bounded basis. More precisely, we construct a separable Hilbertian operator space with the $1$-CBAP but with no completely bounded basis. Our construction relies on a theorem of Oikhberg that provides a structural description of the completely bounded maps on a suitable operator space.

We propose a notion of relative coarse rigidity for $\sigma$-finite von Neumann algebras through modular derivations and their associated GNS-symmetric quantum Markov semigroups, extending Peterson's framework of $L^2$-rigidity to the nontracial setting. Our main result shows that the absence of relatively amenable summands forces the associated deformations to converge on relative commutants in ultrapowers. For diffuse von Neumann algebras with separable predual, this yields a dichotomy between the presence of an amenable summand and rigidity in terms of central sequences. These results give a novel unified approach for proving indecomposability properties such as fullness, relative $\omega$-solidity and relative stable solidity. Our methods apply to examples such as amalgamated free products, operator-valued free Araki-Woods factors and cocycle crossed products.

Let $\Omega=G/\mathbb K$ be an irreducible bounded symmetric domain of rank $r$ and dimension $d.$ In this paper, we study the joint subnormality of $d$-tuple of multiplication operators induced by the coordinate functions on reproducing kernel Hilbert spaces of holomorphic functions on $\Omega$ determined by $\mathbb K$-invariant kernels. We introduce the notion of contractive $d$-tuple associated with $\Omega$ and prove that the weighted Bergman shifts on $\Omega$ are contractive $d$-tuple precisely when they are jointly subnormal. A characterization of joint subnormality further leads to the study of a class of moment problems, which we refer to as twisted moment problems. We establish that the twisted moment problem is equivalent to an appropriate Hausdorff moment problem. This equivalence provides a moment-theoretic characterization of joint subnormality for the multiplication operators under consideration.

We reexamine F. Wiener's classical proof that the Bohr radius of $H^\infty$ is $1/3$. We replace $H^\infty$ by the multiplier algebras of complete Pick spaces and describe the Bohr set for a wide class of such algebras. In particular, we show that the Bohr set of the multiplier algebra of the Drury–Arveson space $H^2_d$ is the closed $\ell^2$-ball of radius $1/3$. Other cases covered by our description of Bohr sets include Dirichlet-type spaces of analytic functions in the unit disc and a class of Hilbert spaces of ordinary Dirichlet series, studied earlier by McCarthy and Shalit.

Dixmier asked in 1967 whether the spectra of simple separable non-type-I $C^*$-algebras all have the same Borel structure. We solve the formulation using ordinary isomorphisms of quotient Borel $\sigma$-algebras by proving that its universal assertion is independent of ZFC, relative to the consistency of ZFC. Under the continuum hypothesis, all separable non-type-I spectra are measurably isomorphic, without simplicity or unitality assumptions. In the model obtained by iterating Sacks forcing for $\omega_2$ stages with countable supports over a GCH ground model, a fixed simple unital universal spectrum is not measurably isomorphic to the CAR spectrum. Neither consistency direction requires a large cardinal. The Borel-definable formulation, which requires isomorphisms to admit Borel lifts between irreducible representations, has a negative answer in ZFC. For the CAR algebra $A$ and a simple separable exact crossed product $B$ associated to $\mathrm{SL}_3(\mathbb{Z})$, there is no Borel-definable injection $\widehat B\to\widehat A$. Nevertheless, all separable nuclear non-type-I spectra are Borel-definably isomorphic. We also determine the answer under several choiceless hypotheses. The axiom of determinacy together with dependent choice gives a stronger negative conclusion: the spectrum of $B$ does not even inject as a set into the spectrum of $A$. The symmetric Solovay model gives the same conclusion. Over ZF, a wellordering of the reals of order type $\omega_1$ suffices for a positive answer. The finite-permutation Feferman–Levy model gives another positive answer for the full sigma-generated Borel structures. These comparisons concern the quotient structures themselves, without measure completion.

We study Lipschitz mappings associated with group actions by similarities, extending the orbit-quotient approach for point-preserving isometric actions. Normalizing by the similarity ratios we obtain isometric representations whose fixed points are character-equivariant Lipschitz mappings. The corresponding quotients by closed spans of orbit differences provide canonical preduals and norm-preserving linearizations, without an amenability assumption. Applications include positively homogeneous, linear, and bilinear mappings. In particular, we prove that the space of positively homogeneous Lipschitz functions form a $1$-complemented subspace of the Lipschitz space (strenghthening the earlier existed result) and interpret existing projections onto linear and bilinear mappings as invariant-mean averages. For amenable groups, the same approach yields isometric realizations of the quotient spaces in the biduals of the original linearization spaces.

We analyse Schrödinger operators with accretive potentials in weighted spaces. We find conditions on potentials and weights for which the Dirichlet realisation, introduced by generalised form methods, has non-empty resolvent set. We establish a domain and graph norm separation property, as well as sufficient conditions for the compactness and Schatten class of the resolvent. Moreover, we investigate the relation between discrete spectra and eigenfunctions of operators in standard and weighted spaces. As applications we extend results on the completeness of eigensystems of operators with accretive potentials from standard to weighted spaces and analyse operator matrices exhibiting a Schur dominance property, in particular, related to a wave equation with strong accretive damping.

We establish exact thresholds for the failure of simultaneous quantum typicality: three parties for mixed states and four for pure states. For the four-qubit Higuchi–Sudbery state, every sequence satisfying the expected purity bounds on three overlapping marginals, with sufficiently small fixed positive entropy slack, is asymptotically perfectly distinguishable from the tensor powers of the input: its trace distance from them tends to the maximum value $2$. This maximal separation persists, with a common positive slack, throughout an open neighborhood in the full four-qubit state space. Tracing out one party yields a mixed tripartite example. The proof connects simultaneous quantum smoothing with polynomial concentration on $\CP^1$. Symmetry reduces the problem to a frame inequality on $\Sym^n(\mathbb C^2)$, where a linear entropy deficit competes with a quadratic cost of polynomial concentration. Bernstein–Markov estimates and a subharmonic barrier control this cost, yielding an exponential obstruction to simultaneous typicality. These results disprove Dutil's multiparty quantum typicality conjecture, formulated in 2011, and the one-shot simultaneous min-entropy-smoothing conjectures of Drescher–Fawzi and Colomer–Winter. Together with the positive results for smaller systems, they determine the smallest numbers of parties at which simultaneous typicality can fail.

We study Yang's higher numerical invariants for the quasi-homogeneous submodules \[ M_{k,\ell}=[z^k-w^\ell]\subset H^2(\mathbb D^2), \qquad k,\ell\in\mathbb N,\quad k\neq\ell. \] The Hilbert–Schmidt property and the low-order invariants $\Sigma_0$ and $\Sigma_1$ for this family follow from earlier results on $M_{\theta,\varphi}$-type submodules. Our aim is to determine the complete higher-order sequence. Using the weighted homogeneous decomposition associated with $z^k-w^\ell$, we construct explicit orthonormal bases for the two defect spaces and determine the exact support of the shifted defect inner products. For $j\geq1$, set \[ A_j=\left\lceil\frac{j}{\ell}\right\rceil, \qquad B_j=\left\lceil\frac{j}{k}\right\rceil. \] We obtain \[ \Sigma_0(M_{k,\ell})=\frac{\pi^2}{6}, \qquad \Sigma_j(M_{k,\ell})=F(A_j,B_j), \] where $F$ is an explicitly evaluated symmetric function on $\mathbb N^2$. The sequence is nonincreasing, tends to zero, and satisfies $\Sigma_j>\Sigma_{j+1} \Longleftrightarrow k\mid j\ \text{or}\ \ell\mid j$. Hence its strict descent set is $k\mathbb N\cup\ell\mathbb N$. This descent set determines the additive semigroup $\langle k,\ell\rangle$, while the complete numerical invariant sequence determines the unordered pair $\{k,\ell\}$. The order of the two exponents is not recoverable because of coordinate-swap symmetry.

We investigate and characterize order automorphisms on the set of partial isometries in the finite-dimensional matrix algebra $M_n(\mathbb{C})$. Different from the classical order automorphisms of subspace lattices, which can be implemented by standard invertible or unitary transformations, the order automorphisms considered herein admit no such conventional matrix representations. Instead, they are essentially governed by matrices such that $I-(A+A^*)$ is either positive or negative invertible. The results reveal that the structural features of order automorphisms for partial isometries are substantially more intricate than those of classical subspace automorphisms.