Papers

In 1981, Bourgain, Rosenthal, and Schechtman constructed uncountably many pairwise non-isomorphic complemented subspaces of $L_p$ for $1 < p < \infty$, $p \neq 2$. We extend their construction to rearrangement-invariant spaces and Hardy spaces. For any separable rearrangement-invariant space $X$ with non-trivial Boyd indices, we show that these subspaces are complemented, previously obtained by Ghawadrah via a different argument, and establish the following dichotomy: Either one can extract an uncountable family of pairwise non-isomorphic subspaces, or they are all isomorphic to $X$. To this end, we introduce an ordinal index that detects whether $X$ embeds complementably into an arbitrary separable Banach space. On the other hand, we give sufficient conditions which imply that in certain spaces close to $L_1$ and $H^1$, these subspaces fail to be complemented.

We study the Arens regularity of the Fremlin projective tensor product of Banach lattice algebras. We firrst introduce and investigate L-weakly almost periodic functionals, a class of functionals defind in terms of L-weak compactness of their orbits. We obtain a characterization of these functionals in terms of bounded disjoint sequences and use it to analyze the Fremlin projective tensor product. In particular, we prove that, for nontrivial Banach attice algebras A and B, Arens regularity of A j jB implies Arens regularity of both A and B. The converse, however, fails: we construct two Arens regular Banach lattice algebras whose Fremlin projective tensor product is not Arens regular. We also establish several sufficient conditions for the Arens regularity of A j jB, including conditions involving the positive Schur property and order continuity of the norm on the space of regular operators. Finally, we show that when both factors are Banach f-algebras, their Fremlin projective tensor product is Arens regular.

In the context of Fell bundles over discrete groups, we introduce and analyze suitable notions of direct sum and tensor product of actions of Fell bundles on Hilbert bundles. For $C^*$-dynamical systems, where actions arise from equivariant representations, we discuss how the tensor product of two such actions is related to the action associated with the tensor product of the underlying equivariant representations. Likewise, since any action gives rise to a $C^*$-correspondence between the full cross-sectional $C^*$-algebras of the involved Fell bundles, we examine the connection between the $C^*$-correspondence associated with the tensor product of two given actions and the tensor product of the associated $C^*$-correspondences. We also discuss how the tensor product of actions naturally appears in Rieffel-like induction processes. Finally we present related results on the Fourier-Stieltjes and Fourier algebras of a Fell bundle.

We establish the following novel heterovariant Gaussian Brunn-Minkowski inequality (involving general covariance matrices $\Sigma_1,\Sigma_2,\Gamma$) for origin-symmetric convex sets $K,L \subset \mathbb{R}^n$ and $p,\lambda \in (0,1)$: \[ \gamma_{\Sigma_1}(K)^{1-p} \gamma_{\Sigma_2}(L)^p \le \left( \left( \frac{1-p}{1-\lambda} \right)^{1-p} \left( \frac{p}{\lambda} \right)^p \right)^{n} \gamma_{\Gamma}((1-\lambda) K + \lambda L) , \] whenever $\Gamma \leq \Sigma_1 \sharp_p \Sigma_2$, the $p$-geometric mean of $\Sigma_1,\Sigma_2$. This immediately implies both the dimensional Gaussian Brunn-Minkowski inequality of Eskenazis-Moschidis (by using $\Sigma_1 = \Sigma_2 = \Gamma$ and optimizing over $p$) and the strong Gaussian (B)-theorem of Cordero-Erausquin-Fradelizi-Maurey (by using $K=L$, $p=\lambda$ and diagonal covariances), fusing them into a single unified statement. In fact, the inequality applies to convex sets whose appropriate Gaussian barycenter is at the origin. Furthermore, it reveals a new *bilinear* version of the strong Gaussian (B)-theorem (involving *two* convex sets); in particular, for all $a,b > 0$ and $\lambda \in [0,1]$: \[ \gamma(a K)^{1-\lambda} \gamma(b L)^{\lambda} \leq \gamma \left ( a^{1-\lambda} b^\lambda \left ((1-\lambda) K + \lambda L \right) \right ) . \] A more general heterovariant version of the (bilinear) (B)-theorem replaces dilation by general linear transformations. A different version yields a heterovariant form of the recently established Gaussian conjugate Rogers-Shephard inequality. By averaging these Gaussian inequalities with respect to the covariance matrices, we obtain dimensional Brunn-Minkowski and conjugate Rogers-Shephard inequalities for Gaussian mixtures. The proofs are based on the Gaussian Forward-Reverse Brascamp-Lieb inequality established in our previous work.

Following \cite{BR} we study hyponormal operators in the strictly larger class $\beta(H)$ of operators whose restriction to every non-zero reducing subspace is norm attaining. The Weyl-type characterization of positive operators via the essential spectrum used in the absolutely norm attaining case is replaced by a well-ordering condition on the eigenvalues (Theorem~\ref{thm:positive}). We prove a transfinite, block-diagonal representation $T=\bigoplus_i\lambda_iS_i$ (with isometries $S_i$) for the $\beta(H)$-operators whose $|T|$-eigenspaces are reducing, equivalently the quasinormal operators with $|T|\in\beta(H)$, and we show by examples that hyponormality alone does not give such a decomposition. For normal operators we obtain a complete representation, and for hyponormal $\beta(H)$-operators we show that normality is equivalent to the spectrum having planar Lebesgue measure zero. An analogous area-zero criterion for normality is proved for hyponormal operators attaining the minimum modulus on every non-zero reducing subspace.

We establish a vector-valued extension of a theorem of Marciszewski. Let $K$ be a compact Hausdorff space, let $\Gamma$ be a set, and let $X$ be a nonzero Banach space containing no isomorphic copy of $c_0$. We prove that if $C(K)$ embeds isomorphically into $c_0(\Gamma,X)$, then $C(K)$ is isomorphic to $c_0(\Lambda)$ for some set $\Lambda$. Moreover, when $\Gamma$ is infinite, the optimal distortion of embeddings of $C(K)$ into $c_0(\Gamma,X)$ equals that of embeddings into $c_0(\Gamma)$. The classification result answers a question posed in an earlier paper.

We establish a spectral theorem for compact self-adjoint linear relations on a Hilbert space. The proof relies on the Arens decomposition of a closed linear relation into an operator part and a purely multivalued part, reducing the problem directly to the classical spectral theorem for compact self-adjoint operators. No finite-dimensionality condition is imposed on the multivalued part. We also prove a finiterank approximation theorem for compact bounded linear relations and a Fredholm alternative for compact self-adjoint linear relations. As an application, we employ the theory of boundary triples to show that the inverse of the Neumann Laplacian on a bounded domain is a compact self-adjoint linear relation. The abstract spectral theorem then recovers the classical eigenfunction expansion of solutions to the Neumann problem: an L2-convergent series in terms of the non-constant Neumann eigenfunctions together with an arbitrary additive constant. The framework extends naturally to general elliptic boundary value problems with finite-dimensional kernel.

We show that if a Banach space $X$ is smooth and almost locally uniformly rotund and if satisfies the following \[0\in W(T) \implies 0\in W(T^{-1})\] for all invertible bounded linear map on $X$ then Birkhoff-James orthogonality in $X$ is symmetric. This result, therefore, characterizes the inner product spaces out of all smooth and almost locally uniformly rotund spaces having three or more dimension via simultaneous zero inclusion property, introduced by Bračič et al. [J. Math. Anal. Appl., 449 (2017) 1413-1423]. Further, this gives answers to some open questions asked by E. Poon [J. Math. Anal. Appl., 534(2024), 128052].

We provide a strengthened answer to the questions posed by Alonso, Martín, and Papini [Studia Math. 188 (2008), 135–150] concerning the comparison between the von Neumann–Jordan and James constants . The classical inequality $\CNJ(X)\leq J(X)$ between the von Neumann–Jordan and James constants was established by Takahashi and Kato [J. Math. Anal. Appl. 359 (2009), 602–609], Wang [Proc. Amer. Math. Soc. 138 (2010), 695–701], and Yang and Li [Appl. Math. Lett. 23 (2010), 277–281], while related sharp estimates were obtained by Kato and Takahashi [Rend. Circ. Mat. Palermo (2) Suppl. 82 (2010), 75–91]. We sharpen this inequality to $\CNJ(X)\leq\JB(X)$, where $\JB(X)$ is the Birkhoff–James orthogonal James constant introduced by Baronti and Papini [Constr. Math. Anal. 5 (2022), 37–45]. Our proof employs optimal distance ellipses and alternating contact directions.

We characterize compactness of the Hilbert-Schmidt semigroups induced by finite-mode Gaussian quantum Markov semigroups with a faithful normal invariant state. A unitary second-quantization representation for every $ s $-embedding shows that compactness at one positive time, compact resolvent, drift stability, and uniqueness of the normal invariant state are equivalent. These conditions imply immediate trace-class membership for all embeddings. We derive exact trace-norm and Hilbert–Schmidt norm formulas, an embedding-independent trace identity, and the generator spectrum with algebraic multiplicities. Examples show that compactness does not require the existence of a spectral gap and that a spectral gap need not imply compactness when the invariant state is not unique.

For $0\leq\alpha<n$, let $A_{*}^{\alpha}$ denote the fractional spherical maximal operator. In this paper we characterise the range of $\beta$ for which the $L^p-L^q$ power-weight estimate \[\left\|A_{*}^{\alpha} f\right\|_{L^q(|x|^{q\beta})} \leq C \left\|f\right\|_{L^p(|x|^{p\beta})}\] holds. In the diagonal case $p=q$, our work gives an affirmative answer to a conjecture of Lee. We prove analogous results for the lacunary fractional spherical maximal operator.

We investigate max-phase isometries on Schatten quasi-classes in the range $0<p<1$. Motivated by the rigidity theory for smooth normed spaces [Results Math. 80 (2025), Article 126], we establish its operator-theoretic counterpart in the quasi-Banach setting. We prove that every surjective max-phase isometry on a finite-dimensional Schatten class is a standard real-linear Schatten isometry multiplied by an even sign-valued function, without any continuity assumption. This extends the scope of the classical Schatten isometry classification [Proc. Edinburgh Math. Soc. 37 (1994), 239–242] to nonlinear mappings preserving only unsigned two-point metric data. Our approach combines support orthogonality, a symmetrized rank-one spectral formula, and a max-profile uniqueness principle. For arbitrary Hilbert spaces, we obtain the corresponding classification on finite-rank operators, while continuity yields the full-space result. We also establish preservation of rank, singular values, and two-sided support orthogonality.

A bipartite observable is stable when its idealized responses are continuous. We show that it is NIP exactly when they are of the first Baire class. They then form a Rosenthal compactum, and the dividing lines of Shelah in model theory, together with the classification of Rosenthal compacta of Todorčević, acquire a physical meaning. For observables, mixtures of states make every hereditarily separable compactum of idealized responses metrizable, and the Todorčević trichotomy becomes a classification into three classes. Each class contains a physical order: the comparison of photon numbers, the comparison of positions, and the causal order of spacetime, whose idealized responses are not first countable because a light cone can shrink to a point; the comparison of two phases on a circle joins it there. Each of the seven minimal families of the heptachotomy, which Argyros, Dodos and Kanellopoulos extracted from the methods of Todorčević, is realized by a tree of preparations of photons, of a particle in a box, or of a spin chain read by a probe; the spin chain also shows that NIP, unlike its uniform version, is not symmetric between the two parties. Finally, for the Ising model in a transverse field we show that, in the ordered phase, the responses of the two pure phases form split pairs whose midpoints, the responses of the symmetric state, form an uncountable discrete set; the proof uses a Lee–Yang theorem for this model.

In 1970, P. R. Halmos raised ten open problems on operator theory. The third problem of Halmos is the following: if an intransitive operator has an inverse, is its inverse also intransitive? This question is reduced to the case of conjugate bi-geometric operators in arXiv: 2610.11202. In the framework of conjugate bi-geometric operators, using tools from complex geometry, operator theory, and function theory, this paper gives an affirmative answer to the third problem of Halmos in the following cases: (a) bundle-preserving operator;(b) subexponential growth of operator norms. These two cases cover a large class of operators.

We establish a Carathéodory-type extension theorem for premeasures with values in general separated abelian net convergence groups. Starting from hereditary exhaustivity on the source ring and intrinsic conditions for combining null errors, we construct a countably additive extension on a regulated Boolean sigma-algebra, with full upper and lower regularity in the original convergence. Completeness is required only on the limit-stable hull of the source range. Under closed summation for arbitrary directed rows and finite-response reflection, we identify the regulated domain with the splitting class of Sion's inner and outer processes. For a Dedekind-complete real vector lattice, we show that sequential closed summation on a bounded interval is equivalent to weak sigma-distributivity of the generated band, giving a converge-theoretic interpretation of the concept. Source representations allow us to recover the partially ordered-group extension of Volauf and the finite-variation extension of Kusraev and Malyugin from the general theorem. We then treat finite semivariation by adjoining dominated coordinate variations to an additive lift of the premeasure. The resulting extension is defined on the full semivariation regulator domain and preserves semivariation exactly, without requiring decomposability or the countable sup property. It includes measures of infinite vector variation.

A normalized sequence $(y_n)$ in the dual $E^{*}$ of a Banach space $E$ is called normal if it is weak-star null, and strongly normal if [ G(y_n)=\left{x\in E:\sum_{n=1}^{\infty}|y_n(x)|<\infty\right} ] is dense in $E$. We prove that, within the class of Grothendieck Banach spaces, reflexivity is characterized by the subsequence behavior of normal sequences: a Grothendieck space $E$ is reflexive if and only if every normal sequence in $E^{*}$ contains a strongly normal subsequence. If $E$ is nonreflexive and Grothendieck, then $E^{*}$ contains a normalized weakly null sequence none of whose subsequences is strongly normal; more strongly, it contains a closed hereditarily $\ell_{1}$ subspace $Y$ with a normalized weakly null Schauder basis such that no sequence in $S(Y)$ is strongly normal. We also obtain an exact quotient criterion valid for arbitrary Banach spaces: a normal sequence has a strongly normal subsequence precisely when, for some subsequence $z=(z_n)$, the quotient $E/\bigcap_n\ker z_n$ is separable. The proofs combine a weak-star basic-sequence argument with results of Haydon, Johnson–Maurey–Schechtman and Popov. Concrete realizations are given in $(L_{\infty}(\mu))^{*}$ for nonatomic probability spaces (and, more generally, for $\sigma$-finite spaces with a nonatomic part) and in $\ell_{\infty}^{*}$.

Autonomous small-body proximity operations require continuous-thrust guidance under uncertain gravity, solar-radiation pressure (SRP), and stochastic disturbances. We combine iterative covariance steering (iCS) with $L_1$ distributionally robust adaptive control ($L_1$-DRAC) to provide risk-aware collision avoidance despite mismatch between the nominal planning model and the true dynamics. Iterative covariance steering plans a nominal feedback policy satisfying distributionally robust conditional value-at-risk keep-out constraints, while $L_1$-DRAC compensates for dynamics uncertainty and bounds the residual distributional mismatch through a Wasserstein radius used by the planner. In $\mathbf{1000}$ paired Monte Carlo trials with gravity, SRP, and diffusion mismatch, $L_1$-DRAC adaptive control reduces the keep-out violation rate from $\mathbf{32.7\%}$ to $\mathbf{0.2\%}$ and the physical collision rate from $\mathbf{16.4\%}$ to zero, without actuator saturation.

We study fully nonlinear uniformly elliptic equations that are imposed only when the gradients of solutions lie outside a closed ball and a closed purely one-unrectifiable set that can be unbounded. We prove Harnack inequalities and Hölder regularity estimates for viscosity solutions, with constants independent of the exceptional set. The proofs use the geometry of the contact sets in the cusp and barrier arguments to exclude exceptional gradients. We also provide examples and counterexamples showing the role of pure one-unrectifiability. Materials and proofs are presented in detail to ensure the paper is fully self-contained and accessible to a broad audience.

We study strict inverse systems of locally convex Lie groupoids and Lie algebroids. We show that strict inverse limits of Lie algebroids and Lie groupoids inherit natural Lie structures and establish a natural isomorphism of functors $\varprojlim\circ\operatorname{Lie}\cong\operatorname{Lie}\circ\varprojlim$. We then consider inverse systems of integrable Banach-Lie algebroids and their source-simply connected levelwise integrations. We give conditions under which these integrations form a strict inverse system of Lie groupoids and hence their inverse limit integrates the inverse-limit algebroid. Under additional hypotheses on the source-fiber inverse sequence, we describe the obstruction to source-simply connectedness of the resulting integration in terms of the derived inverse limit $\varprojlim^{1}\pi_2$ of the second homotopy groups of the source fibers. We illustrate the theory with several examples: inverse limits of jet groupoids, current groupoids, and gauge groupoids, including the corresponding inverse-limit Atiyah algebroids and a source-simply connected inverse-limit integration arising from the quaternionic Hopf bundle.

We study the positive Helson form $h_0[a,b]=\sum_{m,n\ge1}\alpha(mn)a_n\overline{b_m}$, $a,b\in c_{00}$, where $c_{00}$ denotes the finitely supported sequences in $\ell^2(\mathbb{N})$, and $\alpha(n)=\int_{\mathbb{R}}n^{-s}d\mu(s)<\infty$ for a finite positive Borel measure $\mu$. Under the Carleman condition $\sum_{k\ge1}\alpha(4^k)^{-1/(2k)}=\infty$, closability of $h_0$ is equivalent to $\sqrt n\alpha(n)\to0$, and hence to $\mu((-\infty,1/2])=0$. For measures concentrated on $(1/2,\infty)$, let $N_0:\ell^2(\mathbb{N})\supset c_{00}\to L^2(\mu)$ be given by $(N_0a)(s)=\sum_{n\ge1}a_nn^{-s}$. We prove that $N:=\overline{N_0}=N_{\max}$, where $N_{\max}$ acts by the same Dirichlet series on the domain $\operatorname{Dom}(N_{\max})=\{a\in\ell^2(\mathbb{N}):\int_{1/2}^{\infty}|\sum_{n\ge1}a_nn^{-s}|^2d\mu(s)<\infty\}$. The closure $h$ of $h_0$ consequently satisfies $\operatorname{Dom}(h)=\operatorname{Dom}(N)$ and $h[a,b]=\langle Na,Nb\rangle_{L^2(\mu)}$, and its associated positive self-adjoint operator is $H_\mu=N^*N$. We further prove that the sequences $a\in c_{00}$ satisfying $\sum_{n\ge1}a_n/\sqrt n=0$ form an operator core for $H_\mu$. The bounded positive Helson matrices $\mathcal H_{\mu_t}$ generated by $d\mu_t(s)=(1-e^{-t(s-1/2)})d\mu(s)$ give a characterization of its operator domain: $g\in\operatorname{Dom}(H_\mu)$ if and only if $\mathcal H_{\mu_t}g$ converges in $\ell^2$ as $t\to\infty$, with limit $H_\mu g$. Moreover, $H_\mu$ is trace class if and only if $\sum_{n\ge1}\alpha(n^2)<\infty$, and for $\alpha\in\ell^2$ it agrees with the maximal operator defined by the matrix $(\alpha(mn))_{m,n\ge1}$.

We identify explicit closed-form and variational formulae for interpolating the exact values and derivatives through order $s$ of a $C^{s,1}$ function $f:\mathbb{R}^d\to\mathbb{R}$ at $N$ distinct points in $[0,1]^d$. The reconstruction satisfies bounds on its global $C^{s,1}$ seminorm and its Lipschitz constant on $[0,1]^d$ that are independent of the sample size $N$, the latter being the sharp Whitney condition. For $s\ge2$, the reconstruction is real analytic away from the data points and definable in the o-minimal structure $\mathbb{R}_{\exp}$. Our nonlinear extension formula coincides with the formula of McShane (1934) for $s=0$ and with the formulae of Le Gruyer and Phan (2015) and Azagra, Le Gruyer, and Mudarra (2018) for $s=1$, and is new for $s\ge2$. For fixed $d$ and $s\ge2$, our closed-form formula is computable away from the data points by a circuit using only elementary unary and binary real operations, with $\mathcal{O}(N)$ gates and $\mathcal{O}(\log N)$ depth. This yields $\mathcal{O}(N)$ storage and $\mathcal{O}(\log N)$ parallel evaluation time. From the supplied coefficients and weights, the circuit can be compiled with $\mathcal{O}(N)$ one-time work in the exact-real word-RAM model. Thus, in this supplied-data setting, our nonlinear construction improves by a logarithmic factor on the $\mathcal{O}(N\log N)$ initialization bound of Fefferman and Klartag (2009), while retaining linear storage and attaining logarithmic query time through parallel evaluation.

For a Hausdorff locally quasi-convex abelian group $G$, let $\C(G)$ be the poset of all Hausdorff locally quasi-convex group topologies on its underlying group having the same continuous characters as $G$. We prove that, whenever $G$ is non-precompact, $\C(G)$ contains an order-isomorphic copy of $(\Pow(\cont),\subseteq)$, where $\cont=2^{\aleph_0}$. The embedding takes values between the Bohr topology and the original topology. Consequently, every infinite discrete abelian group $D$ satisfies $|\C(D)|=\width\C(D)=2^{2^{|D|}}$. The discrete reduction for locally compact abelian groups then yields exact cardinality and width formulas for all such groups; in particular, both invariants equal $2^{\cont}$ for every non-compact $\sigma$-compact locally compact abelian group. These results answer Questions 6.1–6.3 and 6.5–6.7, and the locally compact case of Question 6.4, posed by L.~Außenhofer and D.~Dikranjan in \cite{AD20}. The embedding also applies to non-compact complete metrizable locally quasi-convex groups. A non-compact precompact nuclear group with a unique compatible topology shows that non-compactness alone does not suffice in the nuclear setting. Finally, the compatible poset of $\R^{\N}$ is not order-isomorphic to that of any discrete abelian group.

The following inequality for vectors $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ was considered in [Linear and Multilinear Algebra 74:1779–1798 (2026)]: \[\|\mathbf{v}^p\|\|\mathbf{w}^p\| - \langle\mathbf{v}^p,\mathbf{w}^p\rangle \leq \|\mathbf{v}\|^p\|\mathbf{w}\|^p - \langle\mathbf{v},\mathbf{w}\rangle^p,\] where $\mathbf{v}^p$ is the vector that is obtained by raising each entry of $\mathbf{v}$ to the power $p$. It was shown that this inequality holds for all positive integers $p$, and it was conjectured that it holds for all real $p \geq 2$ when the entries of $\mathbf{v}$ and $\mathbf{w}$ are positive. A formula for the maximum amount by which the inequality's right-hand side can exceed its left-hand side for nonnegative unit vectors was also conjectured. We prove both of these conjectures, and we determine exactly when the inequality holds with equality. We also present a new generalization of this inequality to three or more vectors.

We investigate the differentiation and integration operator defined on resp. between weighted spaces of analytic functions in the special situation when the weight is given in terms of a weight sequence. More precisely, in this case the weight is defined via the associated weight function which is a crucial object in the ultradifferentiable setting and the information can be then expressed purely in terms of the underlying weight sequence. Due to the additional structure this setting is exploited in order to construct (counter-)examples, to compare conditions on weights in the different weighted frameworks, and to study and to understand in a more precise qualitative way required and admissible growth restrictions on the weights.

We study common weighted sampling designs for operator systems that are exactly observable at nonnegative integer times. For positive operators with original frame condition number at most $\kappa$, we construct a common integer design with arbitrarily small relative Gramian error and $O(\log R)$ distinct times up to $R$. The optimal logarithmic counting coefficient for preserving the frame property satisfies $C^*(\kappa)\sim\pi^{-2}\log\kappa$ as $\kappa\to\infty$. Real and integer times give the same infimum, and we determine its asymptotics as $\kappa\downarrow1$. A finite bound on the sampled condition number strictly increases this coefficient. For tight frames, we determine the exact optimum under this constraint and construct an integer design attaining it. Without this constraint, the infimum is never attained. For positive operators, a design valid for every finite $\kappa$ requires counting faster than $\log R$, with arbitrarily slow divergence of the ratio. For unrestricted normal operators and fixed $\kappa>1$, at least linear counting is necessary. Under the logarithmic sector condition $|\arg\lambda|\le M(-\log|\lambda|)$ with fixed $M\ge0$, logarithmic sampling is possible with a sharp leading coefficient. For fixed Borel spectral sets satisfying a radial density condition, we characterize logarithmic sampling and determine the minimum upper counting exponent, attained by an integer design. Counterexamples show that the radial condition cannot be omitted.

We study uniform approximation for operator learning on compact subsets of Banach spaces. We bound the recovery dimensions and the number of network parameters in terms of compact-set complexity, the target gauge, and recovery stability. For bounded linear operators with a separable domain, we characterize gauge approximability by coordinate decompositions with a common vanishing gauge and arbitrarily small additive errors. Their partial sums give a fixed family of nonlinear encoder–decoder pairs. We prove that nonlinear recovery with a common vanishing gauge and a rank bound depending only on the model dimension is equivalent to the uniform approximation property (UAP). We also obtain linear recovery estimates with the same rank bound and an operator version for the linear factors of nonlinear targets. For Barron operators taking values in a Banach space $Y$ of type $p\in(1,2]$ and admitting a Bochner-integrable density, sampling gives the rate $N^{-(1-1/p)}$ for the expected $L^p(\lambda;Y)$ error, where $\lambda$ is a probability measure on the input compact set. For fixed finite-dimensional factors and a class of Hölder maps, we obtain matching upper and lower parameter bounds at fixed sufficiently large depth. Classical sampling and truncation examples provide explicit recovery maps, and a nonlinear integral operator illustrates the parameter bounds.

In this article, we derive various weighted p-numerical radius inequalities using the Schatten p-norm with the generalized Aluthge transform for operators in Hilbert spaces. In addition, the article explores classical numerical radius and norm inequalities for operators. The bounds obtained in this work improve upon several well-known previous results. In addition, we provide examples where we calculate and analyze upper bounds for the weighted numerical radius of different types of matrices using generalized Aluthge transformation.

Let $\mathcal{H}$ be the separable, infinite-dimensional complex Hilbert space. In the first paper of this series, we introduced, by means of the Rosenblum operators, the shift representation operators $K_x=\sum_{n=0}^{\infty}T^nx\otimes e_n$, which relate an operator $T$ with $r(T)<1$ to the unilateral shift $S$ and reveal its hidden analytic structure. In this paper we replace the pair $(H^2,S)$ by the Sobolev disk algebra $R(\mathbb{D})$ and the multiplication operator $M_z$, whose canonical left inverse $B$ plays the role of $S^*$, and study the two-parameter family $K_{x,y}=\sum_{n=0}^{\infty}T^nx\otimes B^{*n}y$. We prove that $T\in B(R(\mathbb{D}))$ with $r(T)<1$ is intransitive whenever there exist a nonzero vector $g$ and a nontrivial $M_z$-invariant manifold $\mathcal{M}$ such that $f(T)g$ has a zero in the closed unit disk for every $f\in\mathcal{M}$; the proof explicitly constructs a nontrivial invariant subspace of $T$. We further introduce the ideal property and show that every intransitive operator on $\mathcal{H}$ is unitarily equivalent to an operator on $R(\mathbb{D})$ with this property, so that the Invariant Subspace Problem for arbitrary operators reduces to a problem about a concrete function algebra. Applications include a characterization of intransitivity through the rationality of the generating function $\sum_{n=0}^{\infty}\langle T^n\xi,\eta\rangle z^n$ and new results on Pearcy's problem. Our main application is Halmos's third problem, which has remained open for more than fifty years: we reduce it to invertible operators with conjugate bi-geometric form, and we prove that $T^{-1}$ is intransitive for every invertible $T$ admitting a nontrivial projection $P$ with $PTP=TP$ and $PT(I-P)$ of rank one. Finally, we exhibit tridiagonal operators that are intransitive and admit infinite decreasing chains of invariant subspaces.

Kucher asked whether $\ell_{\infty}(E)$ is a Grothendieck space for every super-reflexive Banach space $E$. We answer this question affirmatively by proving the stronger result that $\ell_{\infty}(E)$ has Pełczyński's property $(V)$. Combined with the natural dual representation of $\ell_{\infty}(E)$, this yields that every non-weakly compact operator on $\ell_{\infty}(E)$ fixes a copy of $\ell_{\infty}$. The main ingredient is a general property-$(V)$ theorem for function modules over zero-dimensional compact Hausdorff spaces with stalks admitting a common modulus of uniform convexity. The proof is based on Gierz's integral representation of functionals and the weak compactness of the associated canonical measures. In the case of $\ell_{\infty}(E)$, the relevant stalks are ultrapowers of $E$, and super-reflexivity provides the uniform geometric control required by the theorem. We also derive several consequences for operators on $\ell_{\infty}(E)$, including weak compactness into separable spaces, reflexivity of separable quotients, and compactness of operators into Schur spaces.

We construct a quasi-Banach function lattice $X$ on $\mathbb R$ whose quasi-norm is order continuous and invariant under every translation. Nevertheless, $X$ contains a bounded sequence with common compact support which is equicontinuous in $X$ and has no norm-convergent subsequence. This provides a counterexample to Conjectures 2.5 and 2.11 proposed by Brudnyi in \cite{Brudnyi2015}.