Tingley’s problem asks whether every surjective isometry between the unit spheres of two Banach spaces admits an extension to a surjective real linear isometry between the whole spaces. In this paper, we study Tingley’s problem for ℓ ^p –sums (1
Let n⩾ 2 be an integer and let 𝔻 be a division ring. By T_n(𝔻) we denote the algebra of n× n upper triangular matrices over 𝔻 . In this note, we characterize antiderivations ψ on T_n(𝔻) , i.e., additive maps ψ :T_n(𝔻)→ T_n(𝔻) satisfying ψ (AB)=Bψ (A)+ψ (B)A for all A,B∈ T_n(𝔻) , and show that each antiderivation ψ is inner of the form ψ (A)=α (a_11-a_nn)E_1n for all A=(a_ij) ∈ T_n(𝔻) , where α∈𝔻 , with α = 0 whenever 𝔻 is noncommutative and E_ij are the n× n matrix units with one in the (i, j) entry and zero elsewhere. We then deduce from this result a complete structural characterization of antiderivations ψ :T_n(𝔻)→ T_n(𝔻) on rank k matrices, where 1⩽ k⩽ n is a fixed integer. In particular, when k=n⩾ 3 and 𝔻=𝔽_2 , the field of two elements, each ψ takes the form ψ (A)=λ (a_12E_1,n-1+a_n-1,nE_2n)+∑ _i=1^n-1λ _i a_i,i+1E_1n+∑ _i=1^na_iiX_i for all A=(a_ij) ∈ T_n(𝔽_2) , where λ , λ _1, … ,λ _n-1∈𝔽_2 and X_1, … , X_n ∈ T_n(𝔽_2) with X_1+⋯ +X_n=0 . Some applications to related problems on antiderivations and preserver problems on invertible triangular matrices are included.
Let 𝒟_α^h be the weighted harmonic Dirichlet spaces on the unit disk 𝔻 in the complex plane ℂ . For v(z, z)=∑ _k=0^n∑ _j=0^mc_k,jz^kz^j , we prove that the Hankel operator H_v(z,z) is a Hilbert–Schmidt operator on 𝒟_α^h . Furthermore, given an analytic function f on 𝔻 , we obtain a necessary and sufficient condition for the Hankel operator H_f (or H_f) to be a Hilbert–Schmidt operator on the harmonic Dirichlet space which extends the corresponding result in the weighted Bergman space obtained by Arazy et al. (Am J Math 110(6):989–1053, 1988).
The aim of this study is to describe what we think is the maximal definition space of the Cauchy and of the Hilbert transform in the framework of distribution theory.
