Let A A be a complex square matrix, and write its polar decomposition as A = U | A | A=U|A| . For 0 > λ > 1 0>\lambda >1 , the λ \lambda -Aluthge transform of A A is denoted by \[ Δ λ ( A ) = | A | λ U | A | 1 − λ . \Delta _\lambda (A)=|A|^\lambda U|A|^{1-\lambda }. \] In 2007, Huang and Tam conjectured that the Frobenius norm of the self-commutator is contractive under Δ λ \Delta _\lambda : for every 0 > λ > 1 0>\lambda >1 , \[ ‖ [ A ∗ , A ] ‖ F ≥ ‖ [ Δ λ ( A ) ∗ , Δ λ ( A ) ] ‖ F . \|\left [A^*,A\right ]\|_{F} \ \ge \ \|\left [\Delta _\lambda (A)^*,\Delta _\lambda (A)\right ]\|_{F}. \] If this inequality held, then the iterated self-commutator norms \[ { ‖ [ Δ λ m ( A ) ∗ , Δ λ m ( A ) ] ‖ F } m ∈ N \Bigl \{\bigl \|\left [\Delta _\lambda ^{\,m}(A)^*,\Delta _\lambda ^{\,m}(A)\right ]\bigr \|_F\Bigr \}_{m\in \mathbb N} \] would form a nonincreasing sequence and necessarily converge to 0 0 . In this paper we provide a counterexample, thereby disproving the conjecture. We also obtain the quantitative bounds \[ 3 2 ≤ sup A ∈ M n ( C ) , A ∗ A ≠ A A ∗ 0 > λ > 1 ‖ [ Δ λ ( A ) ∗ , Δ λ ( A ) ] ‖ F ‖ [ A ∗ , A ] ‖ F ≤ 2. \sqrt {\frac 32}\ \le \ \sup _{\substack {A\in \mathbb {M}_n(\mathbb {C}),\ A^*A\neq AA^*\\ 0>\lambda >1}} \frac {\|\left [\Delta _\lambda (A)^*,\Delta _\lambda (A)\right ]\|_F}{\|\left [A^*,A\right ]\|_F} \ \le \ 2. \]
A complete classification of continuous S L ( n ) \mathrm {SL}(n) covariant L p L_p -Minkowski valuations on Orlicz spaces is obtained. Consequently, it establishes a characterization of the asymmetric p p -moment body for functions.
We use Fraïssé theory to study the criteria for the existence of a dense or comeager conjugacy class in the automorphism group of a measure on Cantor space. We characterize Akin’s good measures from [Trans. Amer. Math. Soc. 357 (2005), pp. 2681–2722], as a subclass of ultrahomogeneous measures. We determine good measures with rational values on clopen sets whose automorphism group admits a comeager conjugacy class.
With the aim to better understand the intricate geometry of the class of Lipschitz-free p p -spaces F p ( M ) \mathcal {F}_p(\mathcal {M}) when 0 > p > 1 0>p>1 , in this note we study their Banach envelopes and prove that if 0 > p > 1 0>p>1 and M \mathcal {M} is a metric space then the Banach envelope map of F p ( M ) \mathcal {F}_p(\mathcal {M}) is one-to-one, thus solving in the positive a problem raised by the first author and Kalton [Israel J. Math. 170 (2009), pp. 317–335]. This property has important applications to the linear structure of this family of spaces, being the most immediate one that the dual space of F p ( M ) \mathcal {F}_p(\mathcal {M}) separates the points of F p ( M ) \mathcal {F}_p(\mathcal {M}) .
Gasper gave an extension of the Askey–Roy formula for a beta-type integral defined on the unit circle by increasing the parameters and adding a balancing condition. Tarasov–Varchenko gave a multivariable q q -Selberg type generalization of the Askey–Roy formula. In this paper, we derive a further extension of this Tarasov–Varchenko formula by increasing the parameters and adding a balancing condition. This extension includes Gasper’s extended Askey–Roy formula as the one dimensional case. Our extension is very similar to that of the multidimensional Askey–Wilson integral to the multidimensional Nassrallah–Rahman integral. The proof is done by Aomoto’s method, which uses functions called the fundamental invariants of type A.
We describe all of the irreducible polynomial F p S L 2 ( p r ) \mathbb {F}_pSL_2(p^r) -representations which lift to ( Z / p s Z ) S L 2 ( p r ) (\mathbb {Z}/p^s\mathbb {Z})SL_2(p^r) -representations for s > 1 s>1 , observing that they almost never do. We also show that two related indecomposable F p S L 2 ( p r ) \mathbb {F}_p SL_2(p^r) -representations cannot be lifted to Z / p s Z \mathbb {Z}/p^s\mathbb {Z} -representations for s > 1 s>1 .
We study germs of holomorphic maps at the origin which send the real hyperquadric of signature l = 1 l = 1 in C 3 \mathbb {C}^3 into the homogeneous 2-nondegenerate model in C 4 \mathbb {C}^4 whose infinitesimal symmetry algebra has the “submaximal dimension” 16. Our result exhibits an explicit parametrization for all such maps which are rational and transversal to the model. They are classified into five equivalence classes.
In this paper, we prove that the 1-Wasserstein space P 1 ( X ) \mathcal {P}_{1}(\mathcal {X}) is non-Gromov-compactifiable when X \mathcal {X} is an unbounded Polish metric space (separable and complete metric space). This shows that the statement “ P 1 ( X ) \mathcal {P}_1(\mathcal {X}) is Gromov-compactifiable when X \mathcal {X} is Gromov-compactifiable” is false, although the converse statement holds [Trans. Amer. Math. Soc. Ser. B 12 (2025), 1130–1155].
We consider the infinitesimal generator T T of the right-translation semigroup acting on a weighted Lebesgue space on the half line. We provide a detailed characterization of its adjoint operator T ∗ T^* , and we analyze the spectral properties of T T , thereby contributing to the theory of semigroups arising in models with memory effects.
For G G a connected linear algebraic group over a p p -adic field, we show that the action of G ( B d R + ) G(\mathbb {B}^+_{\mathrm {dR}}) on each Schubert cell in the B d R + \mathbb {B}_{\mathrm {dR}}^+ -affine Grassmannian is transitive in the étale topology on affinoid perfectoids, generalizing a result in the reductive case due to Fargues and Scholze.
