Characterizing Hilbert space frames with the subframe property

Lower bounds for Riesz-Fischer maps in rigged Hilbert spaces

This note concerns a further study about Riesz-Fischer maps, already introduced by the author in a recent work, that is a notion that extends to the spaces of distributions the sequences that are known as Riesz-Fischer sequences. In particular it is proved a characterizing inequality that have as consequence the existence of the continuous inverse of the synthesis operator.

Characterizing arbitrarily slow convergence in the method of alternating projections

In 1997, Bauschke, Borwein, and Lewis have stated a trichotomy theorem that characterizes when the convergence of the method of alternating projections can be arbitrarily slow. However, there are two errors in their proof of this theorem. In this note, we show that although one of the errors is critical, the theorem itself is correct. We give a different proof that uses the multiplicative form of the spectral theorem, and the theorem holds in any real or complex Hilbert space, not just in a real Hilbert space.

框架冗余与贝urling密度

Frame redundancy and Beurling density

该研究证明了框架测度函数与贝urling密度的关系，并展示了框架冗余性的量化定义及采样条件的最优性。

关于希尔伯特空间的一些新的度量刻画

On some new metric characterisations of Hilbert spaces

本文综述了用 Lipschitz 映射替代线性有界映射后，巴拿赫空间刻画为希尔伯特空间的同构版本新结果，重点关注由内积诱导等价范数的存在性。

Interaction decomposition for Hilbert spaces

The decomposition into interaction subspaces is an important result for graphical models and plays a central role for results on the linearized marginal problem; similarly the Chaos decomposition plays an important role in statistical physics at the thermodynamic limit and in probability theory in general. We unify and extend both constructions by defining and characterizing decomposable functors from a well-founded poset to the category of Hilbert spaces, completing previous work on decomposable collections of vector subspaces and presheaves.

Generalized Hilbert Operators

If $g$ is an analytic function in the unit disc $\D $ we consider the generalized Hilbert operator $\hg$ defined by {equation*}\label{H-g} \mathcal{H}_g(f)(z)=\int_0^1f(t)g'(tz)\,dt. {equation*} We study these operators acting on classical spaces of analytic functions in $\D $. More precisely, we address the question of characterizing the functions $g$ for which the operator $\hg $ is bounded (compact) on the Hardy spaces $H^p$, on the weighted Bergman spaces $A^p_α$ or on the spaces of Dirichlet type $\mathcal D^p_α$.

Parkable convex sets and finite-dimensional Hilbert spaces

A subset of a convex body $B$ containing the origin in a Euclidean space is {\it parkable in $B$} if it can be translated inside $B$ in such a manner that the translate the origin. We provide characterizations of ellipsoids and of centrally symmetric convex bodies in Euclidean spaces of dimension $\ge 3$ based on the notion of parkability, answering several questions posed by G. Bergman. The techniques used, which are based on characterizations of Hilbert spaces among finite-dimensional Banach spaces in terms of their lattices of subspaces and algebras of endomorphisms, also apply to improve a…

Compressed resolvents of selfadjoint contractive extensions with exit and holomorphic operator-functions associated with them

Contractive selfadjoint extensions of a Hermitian contraction $B$ in a Hilbert space ${\mathfrak H}$ with an exit in some larger Hilbert space ${\mathfrak H}\oplus{\mathcal H}$ are investigated. This leads to a new geometric approach for characterizing analytic properties of holomorphic operator-valued functions of Kreĭn-Ovcharenko type, a class of functions whose study has been recently initiated by the authors. Compressed resolvents of such exit space extensions are also investigated leading to some new connections to transfer functions of passive discrete-time systems and related classes of…

J-fusion frame for Krein spaces

In this article we introduce the notion of $J$-fusion frame for a Krein space $\mathbb{K}$. We relate this new concept with fusion frames for Hilbert spaces and also with $J$-frames for Krein spaces. We also approximate $J$-fusion frame bounds of a $J$-fusion frame by the upper and lower bounds of the synthesis operator. Finally we address the problem of characterizing those bounded linear operators in $\mathbb{K}$ for which the image of $J$-fusion frame is also a $J$-fusion frame.

Hilbert空间算子换位的相邻截面

Adjacent cross-sections of the commutant of Hilbert space operators

利用几乎不变半空间的存在性技术，为上三角算子矩阵T构造相似模型Ť，并通过研究Ť换位中算子的压缩得到线性流形Ŵ，进而分析相邻截面的匹配条件。

Polynomials with no zeros on a face of the bidisk

We present a Hilbert space geometric approach to the problem of characterizing the positive bivariate trigonometric polynomials that can be represented as the square of a two variable polynomial possessing a certain stability requirement, namely no zeros on a face of the bidisk. Two different characterizations are given using a Hilbert space structure naturally associated to the trigonometric polynomial; one is in terms of a certain orthogonal decomposition the Hilbert space must possess called the "split-shift orthogonality condition" and another is an operator theoretic or matrix condition c…

When do triple operator integrals take value in the trace class?

Consider three normal operators $A,B,C$ on separable Hilbert space $\H$ as well as scalar-valued spectral measures $λ_A$ on $σ(A)$, $λ_B$ on $σ(B)$ and $λ_C$ on $σ(C)$. For any $ϕ\in L^\infty(λ_A\times λ_B\times λ_C)$ and any $X,Y\in S^2(\H)$, the space of Hilbert-Schmidt operators on $\H$, we provide a general definition of a triple operator integral $Γ^{A,B,C}(ϕ)(X,Y)$ belonging to $S^2(\H)$ in such a way that $Γ^{A,B,C}(ϕ)$ belongs to the space $B_2(S^2(\H)\times S^2(\H), S^2(\H))$ of bounded bilinear operators on $S^2(\H)$, and the resulting mapping $Γ^{A,B,C}\colon L^\infty(λ_A\times λ_B\…

由可逆加权复合算子产生的通用性

Universality arising from invertible weighted composition operators

本文研究了可逆加权双曲复合算子的通用性，证明了其特征值平移在Hardy和加权Bergman空间上具有通用性，并讨论了Banach空间情形的推论。

正算子的残差加权分解

Residual-Weighted Decomposition of Positive Operators

本文提出一种基于残差加权更新的正算子分解方法，通过单调动力系统和Parseval框架构造，解决正算子的分解与逆问题。

关于复合算子Berezin范围的凸性及相关问题

On the convexity of the Berezin range of composition operators and related questions

本文研究了再生核Hilbert空间上算子Berezin范围的凸性，刻画了Fock空间和Dirichlet空间上复合算子Berezin范围的凸性，并证明了二维空间上酉等价Berezin范围的椭圆范围定理类似结论。

Multipliers and dual operator algebras

In a previous paper we showed how the main theorems characterizing operator algebras and operator modules, fit neatly into the framework of the `noncommutative Shilov boundary', and more particularly via the left multiplier operator algebra of an operator space. As well as giving new characterization theorems, the approach of that paper allowed many of the hypotheses of the earlier theorems to be eliminated. Recent progress of the author with Effros and Zarikian now enables weak*-versions of these characterization theorems. For example, we prove a result analogous to Sakai's famous characteriz…

Weak topology and Opial property in Wasserstein spaces, with applications to Gradient Flows and Proximal Point Algorithms of geodesically convex functionals

In this paper we discuss how to define an appropriate notion of weak topology in the Wasserstein space $(\mathcal{P}_2(H),W_2)$ of Borel probability measures with finite quadratic moment on a separable Hilbert space $H$. We will show that such a topology inherits many features of the usual weak topology in Hilbert spaces, in particular the weak closedness of geodesically convex closed sets and the Opial property characterizing weakly convergent sequences. We apply this notion to the approximation of fixed points for a non-expansive map in a weakly closed subset of $\mathcal{P}_2(H)$ and of min…