Papers

We study finite polyhedral models inside the projectivized space of equivalent norms $\mathcal{N}'(X)$ on a finite-dimensional Banach space $X$, endowed with the logarithmic distortion metric. Given a finite symmetric direction set $E$, we introduce the class of $\alpha E$-norms, defined as Minkowski functionals of symmetric polytopes whose vertices lie on the rays prescribed by $E$. We identify the admissible weights $\alpha$ for which this parametrization is non-redundant and prove that they give a one-to-one parametrization of the corresponding finite direction model $\mathcal{N}(E)$. We then refine this description by introducing complete, symmetric, simplicial $E$-fans, which encode the conical regions on which the associated polyhedral norms are linear. For a fixed fan $\mathcal{F}$, we define the corresponding geometric and coordinate fan models $\mathcal{N}(\mathcal{F})$ and $\mathcal{R}(\mathcal{F})$, and show that they are naturally isomorphic as cones. After quotienting by positive scalar multiplication, the coordinate models become isometric to the corresponding norm models: the logarithmic distortion metric on $\mathcal{N}'(E)$ and $\mathcal{N}'(\mathcal{F})$ is represented exactly by the Hilbert projective metric on the admissible weight spaces. Finally, we prove completeness results for these finite models and show how they provide finite-dimensional polyhedral approximations to the metric geometry of $\mathcal{N}'(X)$.

We prove that every separable infinite-dimensional Fréchet algebra $X$ admits a continuous linear operator $T$ supporting a dense invariant hypercyclic algebra, giving an affirmative answer to a question of Bayart, Costa Jr. and Papathanasiou. In fact, every dense countable-dimensional subalgebra $\mathcal A$ of $X$ can be prescribed as an invariant hypercyclic algebra. When $X$ admits a continuous norm, the operator can additionally be chosen in the form $T=I+K$, where $K$ is nuclear, so that $\mathcal A = \operatorname{span} \operatorname{Orb}(a,T)$ for any prescribed $a\in\mathcal A\backslash\{0\}$.

Let $\{P_j\}$ be a sequence of projections onto the closed subspaces $\mathcal{M}_j$ of a Hilbert space $\mathscr{H}$. Consider an infinite sequence $P_{i_1}, P_{i_2}, \ldots$ with each $P_{i_n} \in \{P_1, P_2, \ldots\}$, possibly repeating in some order or randomly. The question is: Under what conditions does the sequence $\{P_{i_n} \cdots P_{i_2} P_{i_1} x\}_{n=1}^{\infty}$ converge strongly or weakly to $Px$ for every $x \in \mathscr{H}$, where $P$ is the projection onto the intersection $\mathcal{M} = \bigcap_{i=1}^{\infty} \mathcal{M}_i$? In this paper, we present some results concerning random products of countably infinitely many projections $\{P_j\}_{j=1}^{\infty}$ that incorporates the notion of an infinite-periodic function. More precisely, we introduce a new class of functions $\sigma \colon \mathbb{N} \to \mathbb{N}$, called infinite-periodic functions, and rigorously show that the sequence $\{T_n x\}$ defined by \[ T_1 := P_{\sigma(1)}\quad \mbox{and} \quad T_n := P_{\sigma(n)} T_{n-1} \quad \text{for all } n \geq 2, \] for $x \in \mathscr{H}$, converges weakly to $Px$, where $P$ is the projection onto $\bigcap_{j=1}^{\infty} \mathcal{R}(P_j)$. We also provide some technical examples to illustrate our results.

Let $(\mathscr{E}, \langle \cdot, \cdot\rangle)$ be a Hilbert $C^*$-module over a $C^*$-algebra $\mathfrak{A}$. The space of adjointable operators on $\mathscr{E}$ is denoted by $\mathcal{L}\left(\mathscr{E}\right)$. The sets of all states and pure states on $\mathfrak{A}$ are denoted by $\mathcal{S}\left(\mathfrak{A}\right)$ and $\mathcal{P}\left( \mathfrak{A}\right)$, respectively. For $\tau\in\mathcal{S}\left( \mathfrak{A}\right)$, let us define $\mathcal{N}^{\mathscr{E}}_{\tau}:=\left\lbrace x\in\mathscr{E}:\tau\left( \left\langle x,x\right\rangle\right)=0 \right\rbrace$. The Hilbert completion of ${\mathscr{E}}/{\mathcal{N}^{\mathscr{E}}_{\tau}}$ is denoted by $\mathscr{E}_{\tau}$. For $T\in\mathcal{L}(\mathscr{E})$, the operator $T_{\mathscr{E}_\tau}\in\mathbb{B}\left( \mathscr{E}_\tau\right)$, is defined by $T_{\mathscr{E}_\tau}\left(x+\mathcal{N}^{\mathscr{E}}_{\tau}\right)=Tx+\mathcal{N}^{\mathscr{E}}_{\tau}$. In this paper, we show that $\mathscr{E}_{\tau}={\mathscr{E}}/{\mathcal{N}^{\mathscr{E}}_{\tau}}$ when $\mathfrak{A}$ either is a $C^*$-algebra of compact operators or is commutative. We introduce a quantity in the context of Hilbert $C^*$-modules, denoted by $\pi^{\mathscr{E}}_k(\cdot)$ for $k\geq1$. We prove that $\pi^{\mathscr{E}}_k(T)\leq\sup_{\tau\in\mathcal{P}\left( \mathfrak{A}\right)}\left\|T_{\mathscr{E}_\tau}\right\|_{\left(k\right)}\leq{\pi}^{\mathscr{E}^{\sharp}}_k(T_{\mathscr{E}^{\sharp}})$ for every $T\in\mathcal{L}\left(\mathscr{E}\right)$, where the space $\mathscr{E}^{\sharp}$ is constructed as the extension of $\mathscr{E}$ by the embedding of $\mathfrak{A}$ into its enveloping von Neumann algebra $\mathfrak{A}^{**}$.

We show that there exist two separable real Banach spaces $X$ and $Y$ such that, for every $\varepsilon>0$, there is a standard exact $\varepsilon$-isometry $f_\varepsilon: X\to Y$ whose distortion of every distance is between $0$ and $\varepsilon$, whereas no isometric embedding of $X$ into $Y$ exists, even without a linearity assumption. This gives a negative answer to \cite[Problem 1 and Problem 2]{cheng4} of Cheng and Zhou. The construction combines a snowflaked Lipschitz-free space with the Schur-property theorem of Kalton and the isometric linearization theorem of Godefroy and Kalton.

This work initiates a systematic development of conjugation theory within the framework of Hilbert $C^*$-modules, extending classical complex symmetric operator theory beyond the Hilbert space setting. We first explore the foundational structure of $*$-conjugate-automorphisms $\sharp$ on $C^*$-algebras and characterize their existence through isomorphisms with opposite algebras. We show that the set $\mathbb{L}_c(\mathscr{E})$ of all adjointable conjugate-linear operators on a Hilbert $C^*$-module $\mathscr{E}$ possesses the structure of a Hilbert $\mathbb{L}(\mathscr{E})$-module. We also introduce the key notion of $\sharp$-conjugation on a Hilbert $C^*$-module and fully characterize such modules in terms of real forms and real $C^*$-algebras. Moreover, we investigate the geometric properties of $\sharp$-conjugate duals. A central achievement is the development of polar decomposition theory for semiregular linear (conjugate-linear) operators acting on a Hilbert $C^*$-module equipped with a $\sharp$-conjugation $C$, thereby extending the celebrated Garcia–Putinar theorem [Trans. Amer. Math. Soc. 359 (2007), 3913–3931]. As a consequence, we show that any $C$-symmetric unitary operator factors as a product of two $\sharp$-conjugations. A comparative analysis of alternative definitions shows that requiring the naive adjoint condition $\langle Cx,y\rangle = \langle x,Cy\rangle^*$ would force the underlying $C^*$-algebra to be essentially commutative, thus justifying the adopted approach. We supply various examples and counterexamples that extensively illustrate our results.

Building on a construction of Weifeng Yang, we present a simplified counterexample to Rockafellar's sum conjecture on $c_0(\N_0\times\N)$. The central observation is geometric: the domain of a maximally monotone operator can have a nonconvex norm closure. In our example, this gives a short proof that the operator is not of type (FPV), which in turn implies the failure of the sum conjecture. We define an explicit operator using geometric series and provide a detailed proof of its maximal monotonicity. The example was developed with assistance from GPT-6 Astra. The purpose is to simplify and explain the construction, not to claim an independent counterexample mechanism.

Operator learning is formulated on function spaces, but training data are typically available only through finite-dimensional representations. In encoder–decoder architectures, a matrix-valued kernel on the encoded space induces an operator-valued kernel on the original function spaces, and the corresponding reproducing kernel Hilbert spaces are isometrically isomorphic. As the input and output resolutions increase, the induced kernels converge to a limiting kernel, in the sense of operator-norm convergence of their associated integral operators, allowing regularity assumptions to be stated independently of the encoding resolution. For regularized stochastic gradient descent, we establish upper bounds for decreasing and fixed step sizes, separating the encoding and regularization terms from optimization terms of order \(t^{-\theta}\) and \(T^{-\theta'}\), respectively, for any \(\theta,\theta'\in(0,1)\). We further prove lower bounds showing that these encoding-induced terms are generally unavoidable. The analysis is further extended to encoder–decoder neural networks through the limiting neural tangent kernel (NTK), yielding error bounds with an additional finite-width term and polynomial parameter and sample complexity guarantees when the encoding errors decay algebraically. The framework covers matrix-valued kernels constructed from radial and dot product kernels, NTKs arising from wide encoder–decoder neural networks, and encoder–decoder pairs based on Fourier, Legendre polynomial, wavelet, PCA, or pointwise sampling representations.

We prove that balls are the unique minimisers of the Bogovskii constant among bounded connected Lipschitz domains in $\mathbb{R}^n$, $n\geq 2$, resolving Open Problem 2 of Gazzola, Grunau and Sperone (arXiv:2608.25861).

We study the sampling problem for non-decaying signals defined on a locally compact group. The signals are modeled as elements of suitable subspaces of weighted Orlicz and weighted mixed-norm Orlicz spaces, thereby allowing controlled growth at infinity. Our principal focus is on image spaces of idempotent integral operators. Under suitable oscillation estimates on the associated integral kernels, we establish deterministic sampling theorems that guarantee stable reconstruction from pointwise samples. We further prove an average sampling result in the same setting. In addition, we prove random sampling theorems for functions whose norm is essentially concentrated on a compact subset of the group. In particular, we show that, with high probability, sampling is possible from $\mathcal{O}\!\left(\mu(K)\log \mu(K)\right)$ randomly chosen sample points, where $K$ denotes the compact set on which the signal, belonging to Orlicz space, is essentially norm concentrated. In the same spirit, a random sampling result is proved for signals in mixed-norm Orlicz space. As applications of the abstract theory, we obtain sampling theorems for weighted shift-invariant spaces and Orlicz modulation spaces.

In this note, we prove that every primitive KMS-symmetric quantum Markov semigroup on a finite-dimensional matrix algebra satisfies a modified logarithmic Sobolev inequality (MLSI). We also construct a primitive quantum Markov semigroup without KMS symmetry that fails MLSI, and a primitive KMS-symmetric quantum Markov semigroup that fails complete MLSI (CMLSI). The latter construction also provides a non-primitive KMS-symmetric quantum Markov semigroup without MLSI. For the graph-based KMS-symmetric quantum Markov semigroups studied here, we prove MLSI and CMLSI when the underlying graph is connected and has at least three vertices. Finally, we establish CMLSI for a class of primitive bimodule KMS-symmetric quantum Markov semigroups arising from fermionic systems.

We introduce standard-norm generalized gyrovector spaces: generalized gyrovector spaces whose norm-value line has the ordinary real operations and whose fixed scalar maps are continuous. We prove that every bijective gyrometric isometry between nonempty connected open subsets of two such spaces extends uniquely to a global bijective gyrometric isometry. The proof combines gyromidpoints, point reflections, continuation of local isometry germs, and monodromy; joint continuity of scalar multiplication is derived, rather than assumed. The class includes all nonzero real normed spaces and standardized Möbius, Einstein, and proper-velocity models, and is stable under finite and countable $\ell^p$-products. We also obtain bounded mapping-space examples and, under natural continuity assumptions, standard-norm function-space GGVs $C(K,G)$ for compact $K$. These yield non-Hilbertian and hybrid examples. For every unital JB-algebra, the positive invertible cone carries a one-parameter family of standard-norm GGV structures whose gyrometrics are conjugate to Thompson's metric by power maps. The projective positive cone has a parallel family, based on the variation norm modulo the unit, conjugate to Hilbert's projective metric. The projective structures are contractible. Together with global JB-cone isometry classifications, the abstract theorem yields local-to-global extension and classification results for Thompson and Hilbert geometries. Full-rank density matrices and normalized state spaces of finite-dimensional Euclidean Jordan algebras, including the Albert algebra, give concrete projective examples.

In the present paper, we study the existence of isomorphic embeddings from Lorentz function spaces $L_{p_1,q_1}$ into Lorentz ideals $\mathcal{C}_{p_2,q_2}$ in $B(H).$ In particular, we show that, for $p,q\in (0,\infty),$ the quasi-Banach function space $L_{p,q}(0,1)$ isomorphically embeds into the ideal $\mathcal{C}_{p,q}$ if and only if $(p,q)=(2,2).$ This extends several existing results in the literature and answers a question due to Astashkin et. al.

We examine the generalized Hilbert (matrix) operators $$ H_{g,\gamma} : (a_n) \mapsto \sum_{n=1}^{\infty} \bigg(\frac{k}{n}\bigg)^{\gamma} \frac{g_k a_n}{n+k} $$ on the $\ell^p$ spaces, $1<p<\infty$, where $g=(g_n)$ is a sequence and $-1/p<\gamma <1-1/p$. Given a partition of the natural numbers $\bigcup_j I_j = \mathbb N$, we encode $g$ to our investigation via its $\ell^p$-means $G_j$ over the sets $I_j$. We prove that if $I_j$ increases exponentially, the data $(G_j)$ characterizes boundedness, but is insufficient to determine the exact value of the essential norm. Nonetheless, if the growth rate of $(I_j)$ is subexponential, the corresponding data $(G_j)$, given that the tail converge, is sufficient to determine the exact value, in which case we also calculate it. Moreover, it is shown that the boundedness of $(G_j)$ is sufficient, but not necessary (and necessary, but not sufficient) to ensure $H_{g,\gamma}$ is bounded if the underlying partition $(I_j)$ increases subexponentially (and superexponentially, respectively). Using the dual operator, we also prove that the essential norm of $H_{g,\gamma}$ is comparable with $\limsup_j G(j)$ when $(I_j)$ grows exponentially.

We improve the upper and lower bounds for the smallest Lipschitz constant K for which there exists a K-Lipschitz retraction from the unit ball to the unit sphere of the infinite-dimensional Banach space X, for several classical spaces X.

Let $2\leq k<n/2$ and let $(M^n,[g])$ be a smooth, closed, connected, and locally conformally flat Riemannian manifold with a $k$-admissible metric in the conformal class $[g]$. We prove a stability result of the $\sigma_k$-curvature inequality on $M$, in the sense that if equality is almost satisfied for some conformal metric, then its conformal factor is close to a minimizer of the inequality. Closeness is measured quantitatively in terms of Sobolev norms of the conformal factor, namely with respect to the $W^{1,2}$- and the $W^{1,2k}$-norm. In the non-degenerate case, these norms come with optimal exponents $2$ and $2k$, respectively, whereas in general the exponents are $2+\gamma$ and $\max\{2k,2+\gamma\}$ for some $\gamma\geq 0$ originating from a Łojasiewicz inequality. This extends a previous result by Frank and the author from $k=2$ and the sphere to $2\leq k<n/2$ and the full class of manifolds originally considered by Viaclovsky. It also extends a previous result by Engelstein–Neumayer–Spolaor from $k=1$ to the setting of fully non-linear scalar curvatures.

We study the zero sets of the Zak transform $Z_\lambda B_n(x,\nu)$ of B-splines for $\lambda > 0$. Specifically, we provide a full characterization of the zero set for the hat spline for all positive values of the parameter $\lambda$ and for higher order B-splines when $\lambda > 1$. Finally, we apply these results to establish the frame property of integer-oversampled Gabor systems generated by B-splines. In particular, we characterize the frame property for integer-oversampled Gabor systems generated by the hat splines.

In this short note we prove that, given a $\mathbb{C}$-elliptic operator $\mathcal{A}$ and a map $u\in \mathrm{BV}^{\mathcal{A}}(\Omega;V)$ satisfying \( \nabla_{\mathrm{ap}}u\in L^1(\Omega;V\otimes\mathbb{R}^d), \) then $u\in \mathrm{BV}(\Omega;V)$. The result is quantitative and follows from the Korn-type estimate \[ |Du|(\Omega) \leq C_{d,\mathcal{A}}\left( \|\nabla_{\mathrm{ap}}u\|_{\mathrm{L}^1(\Omega)} + |\mathcal{A}u|(\Omega) \right), \] valid for every such $u\in \mathrm{BV}^{\mathcal{A}}(\Omega;V)$. As a consequence, we obtain the characterization \[ \mathrm{BV}^{\mathcal{A}}(\Omega;V)\setminus \mathrm{BV}(\Omega;V) = \left\{ u\in \mathrm{BV}^{\mathcal{A}}(\Omega;V): \nabla_{\mathrm{ap}}u\notin L^1(\Omega;V\otimes\mathbb{R}^d) \right\}. \] Generative AI has been exploited. The usage is detailed in a specific Section.

We study dimension-free Bohnenblust–Hille inequalities for local operators on systems of $K$-level qudits. For the operator space of support at most $d$, uniformly over all tensor-product orthonormal operator bases, we prove a Bohnenblust–Hille inequality with the optimal exponent $2d/(d+1)$ whose asymptotic exponential base in the interaction order is of order $O(\sqrt K)$. Our proof is intrinsic, based on a one-site scalarization and block-transversal decoupling, and does not rely on a scalarization to the cyclic group or on a Remez-type argument. We complement the upper bound by showing that the asymptotic exponential Bohnenblust–Hille base satisfies $cK^{1/4}\le \beta(K)\le C\sqrt K$ with universal constants $c,C>0$. In this sense, the present work may be viewed as continuing the line of work of Slote–Volberg–Zhang from a complementary intrinsic viewpoint. We also revisit their Gell–Mann and Heisenberg–Weyl scalarization procedures. In the Heisenberg–Weyl setting, the prime-dimensional case already yields the optimal interaction exponent, whereas for composite $K$ the scalar total degree used in their reduction leads to a larger exponent. A support-sensitive formulation recovers the optimal interaction exponent for every $K$, while the intrinsic argument gives the stronger dependence on the local dimension at the level of the asymptotic exponential base. As further consequences, we determine the sharp scale, up to factors exponential in $d$ with base depending only on $K$, of coefficient $\ell_1$-normalization and unconditionality on the exact-support spaces. This yields dimension-free coefficient sparsification, including sparse generalized-Pauli approximation in Heisenberg–Weyl coordinates, as well as normalization and query bounds for canonical LCU/qubitization constructions.

We show that maximal abelian C*-subalgebras (masas) of the Calkin algebra (the algebra of all bounded operators on the separable Hilbert space modulo compact operators) may consistently have their densities strictly less than continuum and we describe many isomorphism types of such masas. Specifically, we prove that after adding any number of Cohen reals to a model of CH the algebra $C(K_{\mathcal A})$ of all complex-valued continuous functions on the Stone space $K_{\mathcal A}$ of a Boolean algebra $\mathcal A$ of cardinality $\omega_1$ is $*$-isomorphic to a masa of the Calkin algebra if and only if $\mathcal A$ does not admit a countably generated ultrafilter. Moreover, for every such Boolean algebra we obtain $\omega_2$ pairwise unitarily non-equivalent such masas, none of which has a commutative lift. We also show in ZFC that if a C*-algebra of the form $C(K)$ for any compact Hausdorff $K$ is $*$-isomorphic to a masa of the Calkin algebra, then no point of $K$ may have character smaller than $\mathfrak p$. Therefore, consistently, there may not be any masa of the Calkin algebra of density less than continuum.

We quantify the failure of interpolation between Hardy spaces $H^p(\mathbb T^\infty)$ on the infinite-dimensional torus. For any set $A\subseteq[1,\infty]$ such that $A\setminus\{\infty\}$ is closed in $[1,\infty)$, we show that there exists an operator that is densely defined on $H^p(\mathbb T^\infty)$ for all $1\leq p<\infty$ and weak-$\ast$ densely defined on $H^\infty(\mathbb T^\infty)$ that extends to a bounded linear operator on $H^p(\mathbb T^\infty)$ if and only if $p\in A$.

We prove that minimal-volume hyperplane sections of the regular simplex through its centroid are parallel to a facet. The proof combines variational methods with Fourier-analytic techniques and zero-diminishing arguments to show that every critical normal vector has at most three distinct non-zero coordinates. Analysis of the two- and three-value cases then yields the sharp lower bound.

We construct compact, exponentially stable $C_0$-semigroups $S$ on Hilbert spaces $X$ such that, for every $T>0$, the stochastic convolution $U_g(t)=\int_0^t S(t-s)g(s)\,\mathrm{d}\beta_s $ is unbounded on $[0,T]$ with positive probability for some predictable $g\in L^\infty(\Omega;L^2(0,T;X))$, where $\beta$ is a real Brownian motion. One example is even an analytic semigroup. For the other, the negative generator $A$ has sectorial angle $\pi/2$ and a bounded $H^\infty$-calculus. Our main tool is a necessary condition: for exponentially stable semigroups, an $L^2$-maximal estimate on $\mathbb R_+$ forces a lower square-function estimate for the negative generator. We deduce this implication from a new identity involving the stochastic convolution, the subordinated Poisson semigroup, and a stopped Brownian motion. Combining this condition with Schauder multipliers on a conditional trigonometric basis yields counterexamples to the maximal estimate. An extrapolation argument then produces the integrands with unbounded stochastic convolutions. Finally, an energy-adapted chaining argument gives continuity and maximal estimates for arbitrary $C_0$-semigroups under the stronger condition $g\in L^2([0,T]; L^\infty(\Omega;X))$.

We prove that every infinite compact abelian group $G$ admits a countably infinite proper set $E\subset\widehat G$ that is strongly small-2 but not Riesz. More precisely, $|\alpha|*|\beta|$ is absolutely continuous for all $\alpha,\beta\in M_E(G)$, whereas $M_E(G)$ contains a singular complex measure of total mass one with exact Fourier support $E$ whose convolution square belongs to $L^2(G)$. When the image of the doubling map on $\widehat G$ is infinite, $E$ may moreover be chosen so that $\#(E\cap(\gamma-E))<\infty$ for every $\gamma$ and $E\cap(-E)=\{0\}$. When that image is finite, we reduce the problem to a nonrectangular construction on the Walsh group, for which $E\cap(E+\gamma)$ is finite at every nonzero shift.

We construct equivalent norms on $L_1[0,1]$, arbitrarily close to the usual $L_1$-norm, which are weakly midpoint locally uniformly rotund and Gâteaux smooth, and whose dual norms have the Daugavet property. In particular, we obtain a strictly convex and smooth Banach space with the Daugavet property, thereby solving a longstanding open problem in the area.

Motivated by the wide applications of the quadratic phase integral transform in signal processing, optics, and time frequency analysis, in this paper we develop a new integral transform called the quaternion quadratic phase Dunkl transform (QQPDT). We present a detailed study of its fundamental properties. In particular, we establish continuity, linearity, scaling, modulation, Riemann Lebesgue result, inversion formula, and Parsevals identity. In addition, we estab lish Heisenberg and Donoho-Stark-type uncertainty principles for the QQPDT. Finally, we develop a signal recovery algorithm in the QQPDT domain. The proposed method enables the reconstruction of missing signal information from partial observations under suitable localization conditions.

We prove that the Bohnenblust–Hille constants for Walsh polynomials on the Hamming cube grow at most polynomially in the degree. More precisely, there is an absolute constant $K$ such that every $f :\{-1,1\}^{n} \to \mathbb{C}$ of degree at most $m$ satisfies $$ \left(\sum_{|S|\le m}|\widehat f(S)|^{2m/(m+1)}\right)^{(m+1)/(2m)} \le Km^{27}\|f\|_\infty. $$ The estimate is uniform in the dimension. The exponent $27$ is not optimized.

We study $\mathbb T^m$-invariant Toeplitz operators on weighted Bergman spaces over the unit ball. A natural unitary transformation represents each such operator as a direct sum of restrictions of Toeplitz operators acting on a polyball. Under an $L^\infty$-valued angular continuity assumption, the corresponding symbols extend from $\mathbb N_0^m$ to a norm-continuous family indexed by the maximal ideal space of the $C^*$-algebra generated by $k$-quasi-radial Toeplitz operators. This extension yields a localization framework for Toeplitz operator algebras. As an application, we consider commutative Banach algebras obtained by adjoining Toeplitz operators whose symbols are invariant under mixed unitary and circle actions. We identify their local quotient algebras on finite-coordinate strata and at infinity and use these identifications to describe their maximal ideal spaces and Gelfand transforms explicitly.

Reconstruction from the coefficients of a frame often requires a dual frame, and the canonical dual is obtained by inverting the frame operator, which is rarely available in a closed form. It is therefore natural to ask whether the duality can instead be implemented by an operator of prescribed form. We study this problem in the context of weighted composition operators on Hardy and Bargmann-Fock spaces. We first characterize those weighted composition operators which carry a frame together with a dual of it to another such pair. We then prove that if the image of a frame under a weighted composition operator is itself a dual frame, then such an operator is a positive scalar multiple of the identity and the frame is tight on both Hardy and Bargmann-Fock spaces. Relaxing duality to approximate duality, we show that on the Hardy space, composition is still forced to be the identity under mild regularity for the weight function. Approximate tightness, however, fails completely. On the Bargmann-Fock space, the situation is rigid: such an approximate dual exists precisely when the ratio of the optimal frame bounds does not exceed an explicit sharp threshold determined by the prescribed error.

Let $\Omega\subset\mathbb Z^d$ be finite and let $S\subset\mathbb T^d$ be measurable. We study the discrete Fourier concentration operator \[ T_{\Omega,S}=P_\Omega B_SP_\Omega, \] which describes simultaneous localization to a finite set of spatial indices and a prescribed spectral region. For independent exponents $\gamma,\eta\in(0,1]$, we introduce translation seminorms for $\Omega$ and $S$ and use them to bound the trace defect \[ \operatorname{tr}(T_{\Omega,S}-T_{\Omega,S}^2). \] The proof combines an exact trace-defect identity with dyadic Fourier estimates, leading to three regimes: when $\gamma=\eta$, the contributions from the dyadic scales are of the same order, and their sum produces a logarithmic factor, while for $\gamma\ne\eta$ the sum is controlled by one end of the scale range. For discretizations $$ \Omega_R=(RF)\cap\mathbb Z^d $$ of a bounded measurable set $F\subset\mathbb R^d$ satisfying an upper Minkowski-neighborhood estimate with exponent $\gamma$, we obtain trace-defect bounds of order $R^{d-\gamma}\log R$ in the critical case and $R^{d-\min\{\gamma,\eta\}}$ off the critical line. These estimates yield quantitative eigenvalue-counting and plunge-region bounds and recover the discrete Landau asymptotic. For a box model at $\gamma=\eta=1$, the logarithmic factor is sharp.