Papers

We give a proof of the convexity of the upper bild of a quaternionic matrix, a theorem of So and Thompson. After reduction to $2\times2$ matrices, we introduce real coordinates on a sphere and obtain two representations of the upper bild over the closed unit ball, involving a convex function and a concave function. Minimizing the first and maximizing the second over parameters with a fixed real part yield a convex lower boundary function and a concave upper boundary function, from which convexity follows.

Let $2<p<\infty$. We prove that every infinite-dimensional closed subspace $F$ of a noncommutative $L_p$-space contains a subspace completely isomorphic to $\ell_p$, $C_p$, $R_p$, or $C_p\cap R_p$ and completely complemented in $F$. This answers the question of Junge and Xu on the existence of completely symmetric basic sequences.

For every von Neumann algebra \(\M\), we prove that each bounded sequence in the Haagerup space \(L_1(\M)=\M_*\) admits either a norm-convergent subsequence or a \(2\)-co-lacunary subsequence. This answers an open problem of Parcet and Randrianantoanina \cite[Remark~3.11]{ParcetRandrianantoanina2006} and removes both hyperfiniteness and semifiniteness.

We prove that, under suitable assumptions, the space of order continuous orthogonally additive operators between Riesz spaces is a Riesz space. As a consequence, we give a negative answer to an open problem posed by Mykhaylyuk and Popov concerning the order continuity of the modulus of an order continuous orthogonally additive operator. We also show that, under an appropriate condition on the domain, every order continuous orthogonally additive operator is order bounded, obtaining an orthogonally additive analogue of the corresponding result for linear operators. Moreover, we establish the order boundedness result in ZF for atomic Riesz spaces and in ZF with dependent choice (DC) for Dedekind complete Riesz spaces. Finally, we characterize when every orthogonally additive operator is an abstract Uryson operator.

The pentablock is a domain in $\mathbb C^3$ arising in the study of the $\mu$-synthesis problem and is defined by \[ \mathcal P = \left\{ (a_{21},\operatorname{tr}(A),\det(A)): A=[a_{ij}]_{i,j=1}^2\in M_2(\mathbb C),\ \|A\|<1 \right\}. \] We introduce the Hardy space $H^2(\mathcal P)$ associated with the canonical measure $\mu$ on the distinguished boundary $b\mathcal P$. We identify $H^2(\mathcal P)$ with a closed subspace of $L^2(b\mathcal P,\mu)$ and show that it is a reproducing kernel Hilbert space. We establish a new description of $\mathcal P$ and its closure and obtain a uniqueness set. In particular, we prove that $\mathcal P$ is the interior of the polynomial convex hull of $\Psi(\overline{\mathbb D}^3)$, where \[ \Psi(\xi,\lambda_1,\lambda_2) = \left( \frac{\xi(\lambda_1-\lambda_2)}{2}, \lambda_1+\lambda_2, \lambda_1\lambda_2 \right) \] is the parametrization map for $\mathcal P$. We also show that $H^2(\mathcal P)$ embeds isometrically as a closed subspace of $H^2(\mathbb D^3)$. We then obtain a Brown-Halmos type characterization of Toeplitz operators on $H^2(\mathcal P)$ and a commutant lifting theorem for $\mathcal P$-isometries and their minimal $\mathcal P$-unitary extensions. As consequences, we determine the commutant of the coordinate multipliers, characterize the compact Toeplitz operators and show that the coordinate multipliers have no nontrivial joint reducing subspaces. Finally, we characterize $\mathcal P$-unitaries as images of commuting triples of unitaries under $\Psi$ and present a class of $\mathcal P$-contractions from triples having $\overline{\mathbb D}^3$ as a spectral set.

We explore properties of the Berezin range $B(C_\phi)$ of a composition operator $C_\phi$ on the Hardy space $H^2$, including constraints on its location in the complex plane $\mathbb{C}$, topological properties, shape, symmetry, and convexity. For example, we show that $B(C_\phi) \subseteq \{z\in \mathbb{C}: \Re(z) > 0 \ \text{and}\ |z| \le 1 + |\phi(0)|\}$, that $B(C_\phi)$ is closed if and only if $C_\phi$ is the identity operator, and that $B(C_\phi)$ containa all its boundary points except for $0$ if $C_\phi$ is compact. Addressing questions raised by Cowen and Felder, we show that $C_\phi$ can be a (nondegenerate) disk and characterize when $B(C_\phi)$ is convex if $\phi$ is linear.

We prove that for $k=4$ the quantum uniformity measures are norms, whereas for every $k\geq5$ the triangle inequality fails. This gives a complete answer to an open question of Bu, Gu, and Jaffe. The $k=4$ case is proved using the Cauchy–Schwarz inequality for the quantum uniformity pairing $\langle\cdot,\cdot\rangle_{Q^4}$ introduced here, together with the triangle inequality in a suitable finite-dimensional Bochner $L^8$ space. For $k\geq5$, we construct systems of linear equations over a finite field and derive a contradiction from the structure theory of norming linear systems.

The theory of matrix monotone functions of finite degree depends subtly on the matrix size. Motivated by this phenomenon, we introduce radial convexity of metrics with respect to matrix orders on the cone of positive definite matrices. We establish radial convexity of several natural metrics with respect to the Loewner order and then focus on the Frobenius norm. Using the Daleckii-Krein formula, we relate radial convexity to the positivity of the Hadamard square of the corresponding Loewner matrix. This leads us to conjecture that radial convexity of order $n$ is equivalent to matrix monotonicity of degree $n$, and we prove this equivalence in degree $2$. We further provide evidence in higher dimensions, obtain a characterization in degree $3$, and discuss extensions of our results to arbitrary unitarily invariant norms.

We prove that, for any two integers $2\le m<n$, a bounded operator $A$ on a complex Hilbert space satisfying \[ A^{*k}A^k=(A^*A)^k, \qquad k\in\{m,n\}, \] is quasinormal. More generally, we show that the inequalities \[ A^{*m}A^m\ge(A^*A)^m \quad \text{and} \quad A^{*n}A^n\le(A^*A)^n \] already imply quasinormality. Together with the known single-exponent counterexamples, this completely resolves the reduced Embry problem for bounded operators: the identities $A^{*k}A^k=(A^*A)^k$, $k\in S\subseteq\{1,2,\ldots\}$, characterize quasinormality if and only if $S$ contains at least two distinct integers greater than one. The proof combines Hansen's inequality and the transformer inequality for weighted operator geometric means with an extremal argument for optimal power bounds.

Every Lipschitz map from a closed subset of a real Hilbert space into a real Hilbert space has an absolutely minimising Lipschitz extension: an extension whose Lipschitz constant on every open subset of the complement, bounded or not, equals the Lipschitz constant of its restriction to the boundary of that subset. Neither space need be separable. This answers a question of Naor and Sheffield. For targets of dimension at least two we prove a density theorem on connected domains: every optimal extension of a boundary datum can be approximated, uniformly on compact sets, by extensions whose Lipschitz constant on every nonempty open subset, and on its boundary, equals the constant of the datum. Consequently, in contrast with the scalar case, such a datum has a unique absolutely minimising extension exactly when its optimal extension is unique, and absolutely minimising extensions need not be closed under uniform convergence, even for fixed smooth data on a planar disk. The extensions are built by finite piecewise isometric interpolation: on nested polyhedra with dense union they are piecewise affine and preserve distances in at least two directions on each simplex, a structure described through leaves of 1-Lipschitz maps. The boundary constraints are handled by a midpoint argument for admissible values, and a Kuratowski-Zorn argument removes the separability assumption while preserving previously prescribed values. We also characterise the bounded linear maps that are absolutely minimising extensions of all data determining their operator norm.

We study algebraic relations among functions in star-invariant subspaces. Let $\theta$ be an inner function and suppose that $f,g,h\in K_{\theta}^p$ satisfy $P(f,g,h)=0$, in which $P$ is a homogeneous polynomial and the associated projective plane curve is smooth. Then $\dim\operatorname{span}\{f,g,h\}\leq1$ if either $\operatorname{deg} P\geq 4$ and the boundary spectrum of $\theta$ has logarithmic capacity zero, or $\operatorname{deg} P = 3$ and the boundary spectrum contains at most one point. This settles new cases of a problem of Dyakonov, subsumes the previously known results, and recovers the classical fact that polynomial parametrizations of a smooth projective plane curve of degree at least three are projectively constant.

For $H^1(\mathbb{R}^2)$-orthonormal families $\{u_n\}_{n=1}^N$ with one-body density $\rho=\sum_{n=1}^N |u_n|^2$, we prove the whole-space endpoint Moser–Trudinger inequality $$ \int_{\mathbb{R}^2}\frac{e^{4\pi\rho(x)}-1-4\pi\rho(x)}{\rho(x)}\,dx \le C\left(N-1+\int_{\mathbb{R}^2}\rho(x)\,dx\right), $$ where the integrand on the left-hand side is defined to be zero on $\{x: \rho(x)=0\}$. We develop a heat-semigroup refinement of Rumin's density method to treat both the endpoint coefficient $4\pi$ and the whole-space setting. Our inequality is sharp in two respects: the optimality of the coefficient $4\pi$ and the linear growth in the rank $N$. Moreover, through a semiclassical analysis of the density generated by normalized Dirichlet eigenfunctions, we establish the collective asymptotic behaviors of the quantities appearing in the inequality in various regimes. These semiclassical behaviors show that the rank-dependent term $N-1$ on the right-hand side is indispensable.

For $T>0$, we consider the Benjamin–Ono-type Dirac equation $$ i\partial_t\psi(x,t) + D(-\Delta)^{\frac{1}{2}}\psi(x,t) = D\,G(\psi(x,t)), \qquad (x,t)\in\mathbb{R}^n\times[0,T], $$ where $(-\Delta)^{\frac{1}{2}}$ is the fractional Riesz operator, $D$ is the Dirac operator associated with the Clifford algebra of signature $(0,n)$, and $G(\cdot)$ a scalar-valued quadratic nonlinearity. Under a pointwise anti-commutation condition on $\psi$ and a uniform dyadic regularity assumption on its spatial Fourier profile, we construct a unit-modulus gauge map $u$ by $ u=\exp\bigl(-iw\bigr),$ where $w$ is a solution of $Dw=\psi$. Assuming, in addition, the stated regularity and commutation conditions on $w$, we show that the Plemelj components $u_{\pm}=P_{\pm}u$ solve a coupled Schrödinger system with gauge nonlinearities $P_{\pm}N(\psi,u)$, where $$ N(\psi(x,t),u(x,t)) = -2P_{-}\!\left[ G(\psi(x,t)) \right]u(x,t). $$ We also obtain a priori estimates for the homogeneous Sobolev seminorms $\dot H^\sigma$ of $\psi(\cdot,t)$, with at most exponential growth on $[0,T]$, together with estimates for $P_{\pm}N(\psi(\cdot,t),u(\cdot,t))$. Combining these bounds with the homogeneous and inhomogeneous Strichartz estimates for the Schrödinger system, we obtain, for $n\geq 3$ and every sharp Schrödinger admissible pair $(q,p)$, an a priori Strichartz-type estimate for $\psi$ in $L_T^q\dot W_x^{\sigma,p}$, expressed in terms of the initial datum $\psi(\cdot,0)$ and the a priori controls of $\psi$ in $L_T^\infty H_x^{\sigma+1}$ and $L_T^\infty L_x^\infty$.

This paper introduces and systematically investigates several classes of numerical ranges associated with real-linear operators on Hilbert spaces, including the essential numerical range, maximal numerical range, essential maximal numerical range, $q$-numerical range, and trace numerical range. We establish fundamental structural properties of these sets and characterize their geometric shape. In particular, we analyze how the real-linear nature of the operators influences the geometry and spectral behaviour of the corresponding numerical ranges, thereby extending and enriching the theory of numerical ranges from complex-linear to real-linear operator settings.

We study several function-theoretic aspects of the domains $G_{E(3;3;1,1,1)}$ and $G_{E(3;2;1,2)}$. We introduce and investigate the notions of $\Gamma_{E(3;3;1,1,1)}$-inner functions and $\Gamma_{E(3;2;1,2)}$-inner functions, and establish their relationships with $\Gamma_{E(2;2;1,1)}$-inner functions and $\Gamma_{E(2;1;2)}$-inner functions. We provide explicit constructions and representations of rational $\Gamma_{E(3;3;1,1,1)}$-inner functions and rational $\Gamma_{E(3;2;1,2)}$-inner functions. As an application of these results, we obtain partial solutions to the associated interpolation problems and construct explicit rational inner interpolants for interpolation data lying in suitable subsets of $G_{E(3;3;1,1,1)}$ and $G_{E(3;2;1,2)}$, respectively.

Assuming that $\mathcal{P}(\omega)$ is tightly $\sigma$-filtered, we prove that $C(\omega^*)$ fails the ball fixed point property. This axiom holds under the continuum hypothesis and in the $\aleph_2$-Cohen model. The proof shows that the Stone space of the countable–cocountable algebra $\operatorname{Cc}(\omega_1)$ is a continuous retract of $\omega^*$ and uses the equivalence between tight $\sigma$-filteredness of $\operatorname{Cc}(\omega_1)$ and of $\mathcal{P}(\omega)$. We also prove that the quotient of a tightly $\sigma$-filtered Boolean algebra $B$ with the countable separation property by an $\aleph_1$-generated $P$-ideal is a Boolean retract of $B$.

Let $G$ be a locally compact, second countable group admitting a Reiter sequence, and suppose that $G$ acts on a von Neumann algebra $\mathcal{M}$ equipped with a semifinite trace $\tau$ via trace-preserving automorphisms. In this paper, we prove a maximal inequality for ergodic averages along such Reiter sequences. Our approach relies on approximating the Reiter sequence by a linear combination of characteristic functions of Følner sequences using the layer-cake representation. Furthermore, we establish the bilateral almost uniform convergence of these ergodic averages. As an application, we deduce a non-commutative Wiener-Wintner type theorem along Reiter sequences.

We study order convergence in ordered vector spaces that are not necessarily lattices. In this setting, we consider two generalizations of order convergence on vector lattices, a directed and an undirected variant. We show that these notions agree for the class of pervasive spaces and extend several results concerning atomicity and order convergence in vector lattices to this class. In particular, we prove that a pervasive JB-algebra is associative, which strengthens a result of Sherman for C*-algebras. We also establish a number of new characterizations of pervasive spaces. Additionally, we systematically study a class of order unit spaces associated to Banach spaces, called ice cream spaces. We provide descriptions of both directed and undirected order convergences in these spaces, yielding examples where the two notions differ. We also consider other order-theoretic aspects of ice cream spaces, such as atoms and lattice operations, relating them to the geometry of the underlying Banach spaces. We demonstrate that ice cream spaces exhibit behavior very different from vector lattices and can serve as a rich source of counterexamples in the theory of ordered vector spaces.

One main aim of our paper is to formulate noncommutative versions of the Lebesgue dominated convergence theorem in the setting of quantum integration theory on von Neumann algebras. Our starting point is a reformulation of classical dominated convergence in terms of weak* convergence. Using our recent sequential-continuity theorem, we obtain dominated convergence formulations for sequentially normal states and, under domination by a fixed bounded integrable majorant, for sequentially normal weights. Integrable, possibly unbounded, majorants are treated first in the tracial setting, then for arbitrary von Neumann algebras using Haagerup's L1 space, and finally using normal semifinite weights and their GNS representations. For any faithful normal semifinite weight $\phi$, weak* convergence of norm-bounded sequences is characterized by `integrating against' bounded positive integrable majorants. We also study noncommutative almost uniform and local almost uniform convergence. In particular, for norm-bounded sequences, local $\phi$-bilaterally almost uniform convergence implies weak* convergence and L1 convergence to 0 of $h^{1/2}(x_n-x)h^{1/2}$ for every positive Haagerup L1 density h, without assuming sigma-finiteness. We also prove a dominated convergence theorem in the predual: fixed positive order domination and Petz's almost uniform convergence imply convergence in the predual norm. We establish implications and equivalences among these convergence notions, connecting dominated convergence with tracial and nontracial integration.

We investigate when a Poincaré inequality forces the underlying measure to be doubling, with particular attention to the space \(X\) controlling the oscillation. We identify \(L\log L\) as the sharp threshold: under a natural local growth condition on the measure, doubling is forced precisely when \(L\log L(0,1)\not\hookrightarrow X\). We also obtain a doubling criterion under a weaker geometric assumption when the oscillation norm is sufficiently stronger than \(L\log L\). We show that this threshold is sharp in every fixed Euclidean dimension, using a dimension-free local \(L\log L\)-Poincaré inequality for log-concave measures and non-doubling Gaussian examples. We then turn to infinite-dimensional spaces, where doubling is impossible for the measures under consideration. Nevertheless, the \(L\log L\)-Poincaré inequality continues to hold, and we establish sharp characterizations for infinite product models, Gaussian measures on Hilbert spaces under power-type small-ball estimates, and Wiener measure with the uniform norm.

We show that the spectrum of the Dirichlet-biLaplacian can be quantitatively bounded from below in terms of the inradius only, for a general open set in dimensions $2$ and $3$. For the same dimensions, we also extend this result to fractional powers $s$ of the Laplacian, with $N/2<s<2$. The resulting estimate comes with a constant having the correct asymptotic behavior as $s$ approaches the two endpoints $N/2$ and $2$. The proofs exploit capacitary techniques in the vein of Maz'ya.

We study the boundary regularity of solutions for the tempered fractional Laplacian \[ (-\Delta)^s_\lambda u=f \quad \text{in }\Omega, \qquad u=0 \quad \text{in }\mathbb{R}^n\setminus\Omega, \] where \(0<s<1\), \(\lambda>0\), and \(\Omega\subset\mathbb{R}^n\) is a bounded Lipschitz domain satisfying the exterior ball condition. The tempered operator is defined by \[ (-\Delta)^s_\lambda u(x) = c_{n,s}\operatorname{P.V.} \int_{\mathbb{R}^n} \frac{u(x)-u(y)}{|x-y|^{n+2s}} e^{-\lambda|x-y|}\,dy, \] which was originally proposed in [Koponen, Phys. Rev. E (1995), 1197-1199]. % Exponential tempering preserves the leading singularity of % the kernel but destroys its homogeneity, preventing a direct % application of classical boundary regularity estimates. % Extending these estimates [Ros-Oton and Serra, J. Math. Pures Appl. (2014), 275–302] to the tempered setting therefore % requires overcoming the difficulties caused by the loss % of scale invariance. By decomposing the operator into the classical fractional Laplacian and the correction operator \(\mathcal Q_\lambda^s\), and combining approximation, localization, and iterative arguments with interpolation inequalities, we establish the global estimate \[ \|u\|_{C^s(\mathbb{R}^n)} \le C\|f\|_{L^\infty(\Omega)}. \] For more regular data, we derive interior and boundary-weighted Schauder estimates. In particular, under suitable regularity assumptions on \(f\), these estimates yield \[ |D^\nu u(x)| \le C\,\delta(x)^{s-|\nu|}, \quad |\nu| > 0, \] where \(\nu\) is a multi-index and \(\delta(x)=\operatorname{dist}(x,\partial\Omega)\). These bounds quantify the possible growth of higher-order derivatives near the boundary and provide a regularity foundation for numerical error analysis of the tempered problem.

Classical harmonic analysis is built upon Lebesgue (or more general) measure. Such a framework is remarkably successful for sets of full ambient dimension, but becomes intrinsically insensitive to lower-dimensional geometry: curves, surfaces, and more generally fractal subsets of \(\mathbb R^n\) with zero \(n\)-dimensional Lebesgue measure. In contrast, Hausdorff content provides a dimension-sensitive notion of size for lower-dimensional sets or fractal sets, but harmonic analysis in this setting is still largely limited to maximal operators. In this paper, we establish a weighted two-parameter good-\(\lambda\) inequality under Hausdorff content setting, providing a route toward other general operators from maximal operators over Hausdorff content. To do this, we firstly identify an appropriate reverse Hölder class $\mathcal R\mathcal H_{q,\delta}$ associated with Hausdorff content, and then prove the self-improving property of $\mathcal R\mathcal H_{q,\delta}$ by establishing the fundamental equivalence between reverse Hölder weights and capacitary Muckenhoupt weights. As direct applications, we obtain weighted Fefferman–Stein inequality and some boundedness on general operators over Hausdorff content. The methodology developed in this paper, even when reduced to the Lebesgue measure case, also provide novel insights, which shows that the good-\(\lambda\) principle is not intrinsically tied to an underlying measure, but can extended to a non-additive framework.

Two basic problems in the theory of integral self-affine tiles are to characterize the digit sets that generate translational tiles and to understand whether such tiles admit periodic translational tilings. We give a complete solution to the first problem: for an expanding integer matrix $\mathbf{A}$ and a digit set $\mathcal{D}\subset\mathbb{Z}^n$ with $\#\mathcal{D}=|\det\mathbf{A}|$, the self-affine set $T(\mathbf{A},\mathcal{D})$ is a tile if and only if, for some $N\geq1$, the iterated digit set $\mathcal{D}_{{\mathbf{A}},N}$ is of skew-product form with respect to $\mathbf{A}^N$. As a consequence, every integral self-affine tile admits a fully periodic self-replicating $q$-fold tiling. Moreover, $q$ divides $|T(\mathbf{A},\mathcal{D})|$. In dimension two we prove more: every integral self-affine tile admits a fully periodic tiling. In particular, the periodic tiling conjecture holds for integral self-affine tiles in $\mathbb{R}^2$.

In this paper we deal with the celebrated Gamow liquid drop model, namely, for $m>0$, $N\geq 2$ and $\alpha\in (0,N)$, we study the problem \[ \min\Big\{P(\Omega)+ V_\alpha(\Omega) : \Omega\subset \mathbb{R}^N,\;|\Omega|=m\Big\}\,, \] and focus mostly on the nonexistence of minimizers for large masses. First we show that any possible minimizer in the asymptotic regime must satisfy an optimal thickness property. Eventually, in the simplified setting of sets made up as a tiling of cubes, we show that nonexistence occurs for large masses for the whole range of parameters $N\geq 2$ and $\alpha\in (0,N)$.

We study weighted and Sobolev estimates for two fractional Schrödinger operators on $\mathbb{R}^n$, where $n\geq3$ and $0<s<1$. For the additive operator $L=(-\Delta)^s+V$, with a nonnegative, nontrivial potential $V$ belonging to a suitable reverse Hölder class, $RH_q$, we establish a critical-radius form inequality and decay estimates for the semigroup mass. These yield a heat-kernel bound combining polynomial off-diagonal decay with a fixed positive exponent of decay relative to the fractional critical radius $\rho_s$. A decomposition at this scale gives weighted strong-type estimates for the semigroup maximal operator and weighted Hardy–Littlewood–Sobolev estimates for $L^{-\alpha/2}$, $0<\alpha s<n$, together with the corresponding endpoint weak-type estimates. The admissible weights satisfy explicit growth restrictions determined by the kernel decay, and the fractional integration order is $\alpha s$. We also consider the spectral operator $Q=(-\Delta+V)^s$. Assuming $V\in RH_{q_1}$ with $n/(1+s)\leq q_1<n$ and classical critical radius $\rho_1(x)\geq1$, we prove that $\nabla Q^{-1/2}$ maps $H^{1-s,p}(\mathbb{R}^n)$ into $L^p(\mathbb{R}^n)$ for $1<p\leq p_0$, where $1/p_0=1/q_1-1/n$. By duality, its adjoint maps $L^{p^{'}}(\mathbb{R}^n)$ into $H^{s-1,p^{'}}(\mathbb{R}^n)$ for $p_0^{'}\leq p^{'}<\infty$.

We establish that Rudin's diagonal restriction operator is contractive from the Hardy space $H^p$ of the bidisc to the Bergman space $A^p$ of the disc for $2\leq p<\infty$, and characterize all cases of equality. We extend the result to polydiscs and obtain applications to polynomial point evaluation among others. The proof relies on an explicit formula, deduced from certain Green's formulas, for the defect between the $p$th powers of the two $L^p$ norms.

We use centered differences to construct Chernoff families and derive even asymptotic expansions for their iterated products. Richardson extrapolation is then used to cancel polynomial part and thereby raise the convergence order. As illustrations, we apply the construction to uniformly parabolic, Schrödinger-type second-order operators and to a third-order dispersive operator, obtaining, at extrapolation depth $m$, the rates $ O(n^{-2m}) $ on the Sobolev scales $H^{2m+\delta}$, $H^{4m}$, and $H^{5m}$, respectively, with arbitrary $\delta>0$. In the analytic category, quantitative all-order estimates allow the extrapolation depth to grow with the base resolution and yield the super-fast rates $e^{-cn^2}$ for the two second-order models and $e^{-cn} $ for the third-order model. The constructions are explicit in terms of the coefficients of the underlying differential operators and do not use the target semigroup in their definition. This provides a constructive positive answer to the challenge of super-fast Chernoff approximation formulated by Remizov in Problem 30 of arXiv:2410.00416.

This paper, we introduce and study biatomic systems in Hilbert spaces, a natural generalization of classical atomic systems; this new concept allows us to represent bounded linear operators using two Bessel sequences simultaneously. In addition, we establish equivalent characterizations of these systems, study their relationship with Bessel biframes and demonstrate the stability of the class under natural operations: duality with respect to the adjunct, orthogonal projections, unit operators and concatenation.

Let $1<s\le2$ and let $H_{s}[0,1]$ consist of the real-valued continuous functions on $[0,1]$ whose graphs have Hausdorff dimension $s$. We prove that every finite-dimensional subspace $E\subset C[0,1]$ with $E\setminus\{0\}\subset H_{s}[0,1]$ is contained in a closed complemented subspace isomorphic to $c_0$, all of whose nonzero elements still belong to $H_{s}[0,1]$. This settles a question of Liu, Shi and Zhang. We also obtain the exact criterion: $H_{s}[0,1]$ is $(\alpha,\mathfrak{c})$-spaceable precisely when $\alpha$ is finite. We also show that every admissible finite-dimensional subspace is contained in a dense subspace of Hamel dimension $\mathfrak{c}$ whose nonzero elements have the same prescribed graph dimension. This also recovers maximal pointwise dense-lineability of $H_{s}[0,1]$.