arXiv subjects

Yicen Ma

Publications and source records attributed to Yicen Ma.

5 recordsLinked to original sources

arXiv2026

A General Proof of the Fong-Tsui Conjecture

We present a general proof of the Fong-Tsui conjecture for bounded operators on arbitrary complex Hilbert spaces. Specifically, we show that $|T|\leq|\operatorname{Re}T|$ implies that $T$ is self-adjoint. The argument combines a positive inverse of a Sylvester map with a spectral cutoff determined by the norm of the positive defect $|\operatorname{Re}T|-|T|$. A local vanishing lemma reduces the analysis to the classical squared self-adjointness criterion, while positivity of the defect yields a global norm contradiction. We formulate the argument as an abstract four-operator vanishing principle, without compactness, trace, or separability assumptions. We also establish a quantitative stability estimate: if $|T|\leq|\operatorname{Re}T|+\varepsilon I$ and $0\leq\varepsilon\leq|T|$, then $|\operatorname{Im}T|\leq6|T|^{7/8}\varepsilon^{1/8}$. The constant is independent of the dimension, and the exponent is not claimed to be optimal. Large language models (LLMs) were used to assist with proof development, algebraic calculations, numerical checks, and auditing of the arguments.

math.FA↗

arXiv2026

Affine Anchors and Cylinder Obstructions in the Three-Dimensional Tingley Problem

Let $X$ be a three-dimensional real Banach space and let $f:S_X\to S_Y$ be a surjective isometry. We study the propagation of an affine formula for $f$ on a relatively open part of $S_X$. A finite family of antipodal distance coordinates propagates a linear anchor except at three explicitly described degeneracies: a facet, an open face-star, or a family of chord cones with a common cylindrical kernel. We prove that, if the norm has no fixed-direction cylindrical open cone, every affine open anchor is linear and global. This yields a source-geometric criterion for the Mazur--Ulam property and covers, among other non-strictly-convex examples, the Euclidean double cone. We also give a segment-saturation criterion which proves the property for every prism $Z\oplus_\infty\mathbb R$, where $Z$ is an arbitrary real Banach plane. Finally, in the remaining cylindrical regime, we prove that a chord-saturated same-kernel network cannot be confined to one proper projective quotient arc. We then show that this multi-arc difficulty and the transverse-ruled part of producing an initial affine anchor reach the same final obstruction: upgrading norm calibration on an ambient open cone, together with calibration on a few spherical segments, to pointwise calibration of the sphere map on a two-dimensional open patch.

math.FA↗

arXiv2026

A proof of the Fong--Tsui conjecture

We prove that a bounded operator $T$ on a complex Hilbert space is self-adjoint whenever $|T|\leq|\re T|$, as conjectured by Fong and Tsui \cite{FT}.

math.FA↗

arXiv2026

Vector-Carleson, Calderón, and Poisson-Atomic Criteria for Generalized Hilbert Operators on $H^p$, $p>2$

Let $\mathcal H_g f(z)=\int_0^1 f(t)g'(tz),dt$, and let $2<p<\infty$. Set $t=2p/(p-2)$ and $X_j=2^{-j/p'}Δ_j g'$, where $Δ_j$ is the hard dyadic Taylor projection. We prove that $\mathcal H_g:H^p\to H^p$ is bounded if and only if $h\mapsto(X_jh)_{j\geq0}$ is bounded from $H^t$ to $\ell^t(H^2)$. The square of this embedding norm equals the norm of the positive column operator $b\mapsto\sum_j b_j|X_j|^2$ from $\ell^{p/2}$ to $L^{p/2}$. Coordinate tails yield essential-norm estimates and an exact compactness criterion, while Hardy duality gives an equivalent paraproduct formulation. The criterion is quantitatively invariant under admissible analytic dyadic resolutions and defines a resolution-independent Calderón symbol space equal to the Hilbert-range multiplier space. We construct a bounded noncompact dense-frequency symbol outside the known blockwise sufficient class. We also prove an exact Poisson-atomic testing theorem: finite positive Poisson mixtures recover the full norm, but no fixed atom count suffices. Finally, aggregate probability densities give an intrinsic atomic-complexity formula and a finite-bandwidth testing bound.

math.FA↗

arXiv2026

Rational Bishop determinants and explicit cyclicity criteria

We study finite-fibre determinants for rational Bishop operators and their role in cyclicity for irrational parameters. The paper has two main parts. First, for the constant vector $f=1$, a resultant identity and a discrete Fourier factorization reveal a determinant parity mechanism for general modular orbit order: odd denominators give a nonnegative normalized determinant on the positive fundamental cell, while for even denominators the unique real alternating Fourier mode is the only factor capable of producing a sign-changing zero. We give an explicit example at $(r,q)=(9,16)$ and an analytic infinite family $(r,q)=(3,6n-2)$. Grivaux's zero-free determinant is identified as the consecutive-order subfamily $D_{1,q}$, so these zeros are caused specifically by nonconsecutive modular ordering. Second, we prove an explicit cyclicity criterion that does not require global nondegeneracy or monotonicity of the fibre determinant. A quantitative Remez estimate controls the small-determinant set; cutoff inverses are approximated by endpoint-corrected Fejer polynomials; and an explicit continuity modulus transfers the resulting rational approximants to irrational parameters. This produces a fully explicit continued-fraction gap function for $f=1$ and, more generally, for every polynomial $f$ with $f(0)\ne 0$. The argument also gives the exact degree and leading coefficient of the corresponding polynomial-vector fibre determinants.

math.FA↗