Notes on Tensor Product Measures

Diagonals of injective tensor products of Banach lattices with bases

In this paper, we show that four main diagonal spaces of injective tensor products are pairwise isometrically isomorphic. When E is a Banach lattice, we show that the tensor diagonal of E is a 1-unconditional basic sequence in both the n-fold injective tensor product of E and the n-fold symmetric injective tensor product of E.

Notes on the Chernoff Product Formula

We revise the strong convergent Chernoff product formula and extend it, in a Hilbert space, to convergence in the operator-norm topology. Main results deal with the self-adjoint Chernoff product formula. The nonself-adjoint case concerns the quasi-sectorial contractions.

A note on the structure of expansive matrices in indefinite inner product spaces

A result on the structure of expansive matrices in an indefinite inner product space is derived, which exhibits the largest unitary compression of the matrix.

Birkhoff-James orthogonality in certain tensor products of Banach spaces

In this article, the relationship between Birkhoff-James orthogonality of elementary tensors in certain tensor product spaces with the Birkhoff-James orthogonality of individual elements in their respective spaces is studied.

Almost squareness and strong diameter two property in tensor product spaces

We study almost squareness and the strong diameter two property in the setting of projective (symmetric) tensor product. We prove that almost squareness is preserved by taking projective tensor product, providing non-trivial examples of ASQ projective tensor product spaces. Furthermore, we give sufficient conditions for a projective symmetric tensor product to have the strong diameter two property which extend most of the known results and provide new examples of such spaces with the strong diameter two property.

Almost square and octahedral norms in tensor products of Banach spaces

The aim of this note is to study some geometrical properties like diameter two properties, octahedrality and almost squareness in the setting of (symmetric) tensor product spaces. In particular, we show that the injective tensor product of two octahedral Banach spaces is always octahedral, the injective tensor product of an almost square Banach space with any Banach space is almost square, and the injective symmetric tensor product of an octahedral Banach space is octahedral.

Two Results on Fremlin's Archimedean Riesz Space Tensor Product

In this paper, we characterize when, for any infinite cardinal α, the Fremlin tensor product of two Archimedean Riesz spaces is Dedekind α-complete. We also provide an example of an ideal I in an Archimedean Riesz space E such that the Fremlin tensor product of I with itself is not an ideal in the Fremlin tensor product of E with itself.

On the Tensor product of archimedean $d$-algebras

The aim of this work is to prove that the Riesz tensor product of two archimedean $d$-algebras is itself a $d$-algebra.

About tensor products of Hilbert quasi *-algebras and their representability properties

This note aims to investigate the tensor product of two given Hilbert quasi *-algebras and its properties. The construction proposed in this note turns out to be again a Hilbert quasi *-algebra, thus interesting representability properties studed in \cite{AT} are maintained. Furthermore, if two functionals are representable and continuous respectively on the two Hilbert quasi *-algebras, then so is their tensor product.

T product Tensors Part I: Inequalities

The T product operation between two three order tensors was invented around 2011 and it arises from many applications, such as signal processing, image feature extraction, machine learning, computer vision, and the multiview clustering problem. Although there are many pioneer works about T product tensors, there are no works dedicated to inequalities associated with T product tensors. In this work, we first attempt to build inequalities at the following aspects: (1) trace function nondecreasing and convexity; (2) Golden Thompson inequality for T product tensors; (3) Jensen T product inequality…

$\ast$-g-frames in tensor products of hilbert $C^{\ast}$-modules

In this paper, we study $\ast$-g-frames in tensor products of Hilbert $C^{\ast}$-modules. We show that a tensor product of two $\ast$-g-frames is a $\ast$-g-frames, and we get some result.

On some tensor inequalities based on the t-product

In this work, we investigate the tensor inequalities in the tensor t-product formalism. The inequalities involving tensor power are proved to hold similarly as standard matrix scenarios. We then focus on the tensor norm inequalities. The well-known arithmetic-geometric mean inequality, H{\" o}lder inequality, and Minkowski inequality are generalized to tensors. Furthermore, we obtain some t-eigenvalue inequalities.

关于矩阵乘积时间均匀集中不等式的注记

A note on time-uniform concentration inequality for matrix products

本文通过简单论证，将固定时间界的矩阵乘积集中性界推广到任意时间界，受Oja算法分析启发。

Essential spectra of tensor product Hilbert complexes, and the $\overline\partial$-Neumann problem on product manifolds

We investigate tensor products of Hilbert complexes, in particular the (essential) spectrum of their Laplacians. It is shown that the essential spectrum of the Laplacian associated to the tensor product complex is computable in terms of the spectra of the factors. Applications are given for the $\overline\partial$-Neumann problem on the product of two or more Hermitian manifolds, especially regarding (non-) compactness of the associated $\overline\partial$-Neumann operator.