Trending in Functional Analysis
2602.19599Involutions and angles between subspaces
We provide a complete structure theorem for involutory matrices. This yields a new approach to principal angles between subspaces and provide a series of nice formulae for these angles.
2602.195992
Feb 2026Functional Analysis
2602.17485On the three space property for $C(K)$-spaces
Assuming $\mathfrak p=\mathfrak c$, we show that for every Eberlein compact space $L$ of weight $\mathfrak c$ there exists a short exact sequence $0\to c_0\to X\to C(L)\to 0$, where the Banach space $X$ is not isomorphic to a $C(K)$-space.
2602.174856
Feb 2026Functional Analysis
2602.17432The behaviour of quasi-linear maps on $C(K)$-spaces
In this paper we combine topological and functional analysis methods to prove that a non-locally trivial quasi-linear map defined on a $C(K)$ must be nontrivial on a subspace isomorphic to $c_0$. We conclude the paper with a few examples showing that the result is optimal, and providing an application to the existence of nontrivial twisted sums of $\ell_1$ and $c_0$.
2602.174324
Feb 2026Functional Analysis
2602.14047On the Schur-Agler Norm
We establish a new description of the Schur-Agler norm of a holomorphic function on the polydisc as the solution of a convex optimization problem. Consequences of this description are explored both from a theoretical and from a practical point of view. Firstly, we give unified proofs of the known facts that the Schur-Agler norm can be tested with diagonalizable or nilpotent matrix tuples, as well as a new proof of the existence of Agler decompositions. Secondly, we describe the predual of the Schur-Agler space as a space of analytic functions on the polydisc. Thirdly, we give a unified treatment of existing counterexamples of von Neumann's inequality in our framework, and exhibit several methods for constructing counterexamples. On the practical side, we explain how the Schur-Agler norm of a homogeneous polynomial can be numerically approximated using semidefinite programming.
2602.140471
Feb 2026Functional Analysis
2602.12362Infinite-Dimensional Lie Groups
This is a preliminary version of a book on infinite-dimensional Lie groups. It covers the basics of calculus and manifolds in the context of locally convex spaces, based on Bastiani's notion of a smooth map. Starting from this concept, we develop the basics of smooth manifolds and define Lie groups as manifolds with smooth group operations. We discuss in particular several classes of Lie groups, such as regular ones, or those with an exponential function that is a local diffeomorphism. The local theory, subgroups and quotients are explored in some detail. Classes of Lie groups that are discussed in detail include: unit groups of continuous inverse algebras, groups of smooth maps, direct limit groups and groups of diffeomorphism. We also included chapters on the topology of infinite-dimensional Lie group and on various selected topics.
2602.12362249
Feb 2026Functional Analysis
A new class of positive linear operators preserving logarithmic functions
In this paper, we introduce a new class of positive linear operators that generalize the classical Bernstein operators. Specifically, we construct a sequence of operators that reproduce the logarithmic function $\ln(1+μ+x)$, with $μ> 0$ and $x \in [0,1]$. We prove pointwise and uniform convergence and we derive a quantitative estimate of the approximation error in terms of the modulus of continuity. We also obtain a Voronovskaja-type asymptotic formula, that is used to establish saturation results and inverse theorems. In particular, the saturation class of the considered approximation process is characterized by solving a second order differential equation. Shape-preserving properties, such as monotonicity, concavity and variation diminishing, are also investigated. Finally, a simple application to signal denoising is addressed.
2602.061681
Feb 2026Functional Analysis
Di-Graphs with tightly connected Clusters: Effective Graph Laplacians and Resolvent Convergence
In this note, we study Laplacians on graphs for which connectivity within certain subgraphs tends to infinity. Our main focus are graphs sharing a common node set on which edge weights within certain clusters grow to infinity. As intra-cluster connectivity increases, we show that the corresponding graph Laplacians converge -- in the resolvent sense -- to an effective graph Laplacian. This effective limit Laplacian is defined on a coarsened graph, where each highly connected cluster is collapsed into a single node. In the undirected setting, the effective Laplacian arises naturally from aggregating over tightly connected clusters. In the directed case, the limiting graph structure depends on the precise manner in which connectivity increases; with the corresponding effects mediated by the left and right kernel structure of the Laplacian restricted to high-connectivity clusters. Our results shed light on the emergence of coarse-grained dynamics in large-scale networks and contribute to spectral graph theory of directed graphs.
2601.180571
Jan 2026Functional Analysis
2601.18055Large Coupling Convergence Beyond Definiteness
We study convergence of operator families of the form $A_β= A + βB$ towards an effective operator defined on $\ker(B)$, as the coupling constant $β$ tends to infinity. Crucially, we focus on the setting where neither $A$ nor $B$ can be assumed to be positive- (or negative-) semi-definite. We are hence outside the classical form-theoretic framework, where results based on Kato's monotone convergence theorem would be applicable. Thus, instead of form methods, our approach builds on classical resolvent identities to study convergence of the family $\{A_β\}_β$. Our findings are that: (i) \emph{Strong} resolvent convergence holds (without further spectral assumptions) if $A + βB$ is self-adjoint and the compression of $A$ onto $\ker(B)$ is well behaved. (ii) Under the more detailed assumption that $0 \in σ(B)$ is isolated, \emph{norm} resolvent convergence can be established even if $A+βB$ is merely closed, provided the quasinilpotent part of $B$ at zero vanishes and certain conditions on the interplay of $A$ and $B$ are met. Importantly, if $B$ is not self-adjoint we find that the limit operator not only depends on $\ker(B)$ as a Hilbert space, but crucially also on the precise form of the Riesz projector at $0 \in σ(B)$ onto $\ker(B)$.
2601.180551
Jan 2026Functional Analysis
2601.11934The nonlinear estimates on quantum Besov space
The superposition operators have been widely studied in nonlinear analysis, which are essential for the well-posedness theory of nonlinear equations. In this paper, we investigate the boundedness estimates of superposition operators with non-smooth symbols on quantum Besov spaces, which significantly generalize McDonald's results \cite{McNLE} for infinitely differentiable symbols and have rich applications in the well-posedness theory of noncommutative PDEs. As a byproduct, we prove the equivalence of the two descriptions of quantum Besov spaces, resolving the conjecture proposed in \cite[Remark 3.16]{McNLE}. The new ingredients in the proof also involve quantum chain rule and nonlinear interpolation.
2601.119341
Jan 2026Functional Analysis
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 $\mathcal{L}$.
2601.091552
Jan 2026Functional Analysis
2601.09145A 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.
2601.091453
Jan 2026Functional Analysis
2601.09094A Sharp Localized Weighted Inequality Related to Gagliardo and Sobolev Seminorms and Its Applications
In this article, we establish a nearly sharp localized weighted inequality related to Gagliardo and Sobolev seminorms, respectively, with the sharp $A_1$-weight constant or with the specific $A_p$-weight constant when $p\in (1,\infty)$. As applications, we further obtain a new characterization of Muckenhoupt weights and, in the framework of ball Banach function spaces, an inequality related to Gagliardo and Sobolev seminorms on cubes, a Gagliardo--Nirenberg interpolation inequality, and a Bourgain--Brezis--Mironescu formula. All these obtained results have wide generality and are proved to be (nearly) sharp. The original version of this article was published in [Adv. Math. 481 (2025), Paper No. 110537]. In this revised version, we correct an error appeared in Theorem 1.1 in the case where $p=1$, which was pointed out to us by Emiel Lorist.
2601.090941
Jan 2026Functional Analysis
2601.09015Relative uniform completion of a vector lattice
In the paper, we revisit several approaches to the concept of uniform completion $X^{\mathrm{ru}}$ of a vector lattice $X$. We show that many of this approaches yield the same result. In particular, if $X$ is a sublattice of a uniformly complete vector lattice $Z$ then $X^{\mathrm{ru}}$ may be viewed as the intersection of all uniformly complete sublattices of $Z$ containing $X$. $X^{\mathrm{ru}}$ may also be constructed via a transfinite process of taking uniform adherences in $Z$ with regulators coming from the previous adherences. If, in addition, $X$ is majorizing in $Z$ then $X^{\mathrm{ru}}$ may be viewed as the uniform closure of $X$ in $Z$. We show that $X^{\mathrm{ru}}$ may also be characterized via a universal property: every positive operator from $X$ to a uniformly complete vector lattice extends uniquely to $X^{\mathrm{ru}}$. Moreover, the class of positive operators here may be replaced with several other important classes of operators (e.g., lattice homomorphisms). We also discuss conditions when the uniform adherence of a sublattice equals its uniform closure, and present an example (based on a construction by R.N. Ball and A.W. Hager) where this fails.
2601.090151
Jan 2026Functional Analysis
2601.00283On automatic boundedness of operators between ordered and topological vector spaces
We study topological boundedness of order-to-topology bounded and order-to-topology continuous operators from ordered vector spaces to topological vector spaces. Levi and Lebesgue operators are investigated.
2601.002831
Jan 2026Functional Analysis
2512.24575Functional Calculi, Positivity, and Convolution of Matrices
Convolution admits a natural formulation as a functional operation on matrices. Motivated by the functional and entrywise calculi, this leads to a framework in which convolution defines a matrix transform that preserves positivity. Within this setting, we establish results parallel to the classical theories of Pólya--Szegő, Schoenberg, Rudin, Loewner, and Horn in the context of entrywise calculus. The structure of our transform is governed by a Cayley--Hamilton-type theory valid in commutative rings of characteristic zero, together with a novel polynomial-matrix identity specific to convolution. Beyond these analytic aspects, we uncover an intrinsic connection between convolution and the Bruhat order on the symmetric group, illuminating the combinatorial aspect of this functional operation. This work extends the classical theory of entrywise positivity preservers and operator monotone functions into the convolutional setting.
2512.245751
Dec 2025Functional Analysis
2512.10646Eigenfunctionals for positive operators
We establish an eigenfunctional theorem for positive operators, evocative of the Krein--Rutman theorem. A more general version gives a joint eigenfunctional for commuting operators.
2512.106461
Dec 2025Functional Analysis
2512.08806The Cahill-Casazza-Daubechies problem on Hölder stable phase retrieval
Phase retrieval using a frame for a finite-dimensional Hilbert space is known to always be Lipschitz stable. However, phase retrieval using a frame or a continuous frame for an infinite-dimensional Hilbert space is always unstable. In order to bridge the gap between the finite and infinite dimensional phenomena, Cahill-Casazza-Daubechies (Trans.Amer.Math.Soc. 2016) gave a construction of a family of nonlinear subsets of an infinite-dimensional Hilbert space where phase retrieval could be performed with a Hölder stability estimate. They then posed the question of whether these subsets satisfied Lipschitz stable phase retrieval. We solve this problem both by giving examples which fail Lipschitz stability and by giving examples which satisfy Lipschitz stability.
2512.088062
Dec 2025Functional Analysis
On the existence of large subspaces of $C(K)$ that perform stable phase retrieval
The purpose of this article is to address an open problem posed by Freeman-Oikhberg-Pineau-T.~(\textit{Math.~Ann.}~2024) regarding the existence of large subspaces of $C(K)$ that perform stable phase retrieval (SPR). We begin by proving that for both the real and complex fields, the space $C(K)$ admits an infinite-dimensional SPR subspace if and only if the second Cantor-Bendixson derivative $K{''}$ is nonempty. We then show how to construct ``large" SPR subspaces of $C(K)$, where the size of the subspace depends quantitatively on the number of non-trivial Cantor-Bendixson derivatives that the compact Hausdorff space $K$ possesses.
2512.081142
Dec 2025Functional Analysis
Isometric embeddings into $C(K)$-spaces doing stable phase retrieval
Motivated by a question posed by Freeman, Oikhberg, Pineau and Taylor, we prove that if $K$ is a compact Hausdorff space with $K^{(α)}\neq\varnothing$, where $2<α<ω$, then $C[1,ω^α]$ isometrically embeds into $C(K)$ doing stable phase retrieval (SPR). We also show that the latter cannot be extended to the case $α=2$.
2512.081102
Dec 2025Functional Analysis
2512.05708Arens products for some convolution algebras of measures
We consider the measure algebra of a Chébli-\!Trimèche hypergroup (in particular, double coset spaces of classical Lie groups) and study the corresponding Arens products on its second dual. The behaviour turns out to be different to the group case investigated in [LNPS]. For this, we study more closely properties of the multiplication and generalized translation in a Chébli-\!Trimèche hypergroup and the asymptotic behaviour.
2512.057081
Dec 2025Functional Analysis
