Bin Meng
In this paper, we investigate operator-valued frames with the structure of group-like unitary system. We show the commutant of the group-like unitary system can be characterized in terms of analysis operators associated with all the operator-valued Bessel generators and give some sufficient and necessary conditions to describe when an operator-valued frame generator admits a Parseval dual.
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We establish a characterization of unitary equivalence of two bilateral operator valued weighted shifts with quasi-invertible weights by an operator of diagonal form. We also present an example of unitary equivalence between shifts defined on $\mathbb{C}^2$ which cannot be given by any unitary operator of diagonal form. The paper is concluded with investigation of unitary operators than can give unitary equivalence of bilateral operator valued weighted shifts.
We systematically study various aspects of operator-valued multishifts. Beginning with basic properties, we show that the class of multishifts on the directed Cartesian product of rooted directed trees is contained in that of operator-valued multishifts. Further, we establish circularity, analyticity and wandering subspace property of these multishifts. In the rest part of the paper, we study the function theoretic behaviour of operator-valued multishifts. We determine the bounded point evaluation, reproducing kernel structure and the unitary equivalence of operator-valued multishifts with inv…
Following the work of Semyon Alesker in the scalar valued case and of Thomas Wannerer in the vector valued case, the dimensions of the spaces of continuous translation invariant and unitary equivariant tensor valuations are computed. In addition, a basis in the vector valued case is presented.
Given a purely contractive matrix-valued analytic function $Θ$ on the unit disc $\bm{D}$, we study the $\mc{U} (n)$-parameter family of unitary perturbations of the operator $Z_Θ$ of multiplication by $z$ in the Hilbert space $L^2_Θ$ of $n-$component vector-valued functions on the unit circle $\bm{T}$ which are square integrable with respect to the matrix-valued measure $\Om_Θ$ determined uniquely by $Θ$ and the matrix-valued Herglotz representation theorem. In the case where $Θ$ is an extreme point of the unit ball of bounded $\bm{M}_n$-valued functions we verify that the $\mc{U} (n)$-paramet…
On unitary equivalence and reducing subspaces of analytic Toeplitz operator on vector-valued Hardy space
该研究证明向量值Hardy空间上的$T_{z^n}$算子酉等价于多份标量Hardy空间上$T_z$的直和，并结合矩阵与算子理论方法完整刻画了该算子的约化子空间。
Classical Segal-Bargmann theory studies three Hilbert space unitary isomorphisms that describe the wave-particle duality and the configuration space-phase space. In this work, we generalized these concepts to Clifford algebra-valued functions. We establish the unitary isomorphisms among the space of Clifford algebra-valued square-integrable functions on $\mathbb{R}^n$ with respect to a Gaussian measure, the space of monogenic square-integrable functions on $\mathbb{R}^{n+1}$ with respect to another Gaussian measure and the space of Clifford algebra-valued linear functionals on symmetric tensor…
We consider a refinement of triangular factorization for unitary matrix valued loops.
Unitary parts of Toeplitz operators with operator-valued symbols
受希尔伯特空间压缩算子典范分解启发，研究向量值Hardy空间上压缩Toeplitz算子的非零酉约化子空间存在条件，给出等价刻画并推广已有结果，还得到解析Toeplitz算子的更精细刻画。
The unitary equivalence of $2$-isometric operators satisfying the so-called kernel condition is characterized. It relies on a model for such operators built on operator valued unilateral weighted shifts and on a characterization of the unitary equivalence of operator valued unilateral weighted shifts in a fairly general context. A complete system of unitary invariants for $2$-isometric weighted shifts on rooted directed trees satisfying the kernel condition is provided. It is formulated purely in the langauge of graph-theory, namely in terms of certain generation branching degrees. The members…
Vector valued de Branges spaces, CNU contractions and functional models
本文通过Hilbert空间直和分解构造向量值再生核Hilbert空间，证明其为de Branges空间，并用于建立完全非酉压缩算子的函数模型，同时给出Fredholm型判据并建立特征函数与投影算子值函数的关系。
Quantum Kolmogorov--Arnold representation theorem for continuous unitary-valued maps
研究连续酉值映射的量子柯尔莫哥洛夫 - 阿诺德表示定理，通过反厄米值映射矩阵指数内的精确加法分解及酉映射的有限序列积等方法，给出两个定理并以反例表明局部定理不能全局扩展。
An analog of the Paley-Wiener isomorphism for the Hardy space with an invariant measure over infinite-dimensional unitary groups is described. This allows us to investigate on such space the shift and multiplicative groups, as well as, their generators and intertwining operators. We show applications to the Gauss-Weierstrass semigroups and to the Weyl-Schrödinger irreducible representations of complexified infinite-dimensional Heisenberg groups.
For a fixed natural number n, we consider a family of rank n unitary perturbations of a completely non-unitary contraction (cnu) with deficiency indices (n,n) on a separable Hilbert space. We relate the unitary dilation of such a contraction to its rank n unitary perturbations. Based on this construction, we prove that the spectra of the perturbed operators are purely singular if and only if the operator-valued characteristic function corresponding to the unperturbed operator is inner. In the case where n=1 the latter statement reduces to a well-known result in the theory of rank one perturbat…
The infinitesimal generator $A$ of a strongly continuous semigroup on a Hilbert space is assumed to satisfy that $B_β:=A-β$ is a sectorial operator of angle less than $\fracπ{2}$ for some $β\geq 0$. If $B_β$ is dissipative in some equivalent scalar product then the Naimark-Arocena Representation Theorem is applied to obtain a Kre\uın space unitary dilation of the semigroup.
In this article, we show that, if $S\in \mathcal{B}(H)$ is irreducible and essential unitary, then $\{S,SS^*\}$ is a hyperrigid generator for the unital $C^*$-algebra $\mathcal{T}$ generated by $\{S,SS^*\}$. We prove that, if $T$ is an operator in $\mathcal{B}(H)$ that generates an unital $C^*$-algebra $\mathcal{A}$ then $\{T,T^*T,TT^*\}$ is a hyperrigid generator for $\mathcal{A}$. As a corollary it follows that, if $T\in \mathcal{B}(H)$ is normal then $\{T,TT^*\}$ is hyperrigid generator for the unital $C^*$-algebra generated by $T$ and if $T\in \mathcal{B}(H)$ is unitary then $\{T\}$ is hyp…
A short note on similarity of operator-valued multishifts
本文研究带可逆算子权重的算子值多重移位的相似性，给出完全由算子权重表述的刻画，并将其用于判定对角核再生核 Hilbert 空间上坐标乘法算子元组的相似性。
We prove the existence of a complex valued $C^2$-function on the unit circle, a unitary operator U and a self-adjoint operator Z in the Hilbert-Schmidt class $S^2$, such that the perturbated operator $$ f(e^{iZ}U)-f(U) -\frac{d}{dt}\bigl(f(e^{itZ}U)\bigr)_{\vert t=0} $$ does not belong to the space $S^1$ of trace class operators. This resolves a problem of Peller concerning the validity of the Koplienko-Neidhardt trace formula for unitaries.
We investigate the structured frames for Hilbert $C^{*}$-modules. In the case that the underlying $C^{*}$-algebra is a commutative $W^*$-algebra, we prove that the set of the Parseval frame generators for a unitary operator group can be parameterized by the set of all the unitary operators in the double commutant of the group. Similar result holds for the set of all the general frame generators where the unitary operators are replaced by invertible and adjointable operators. Consequently, the set of all the Parseval frame generators is path-connected. We also obtain the existence and uniquenes…
Consider $d$ commuting $C_{0}$-semigroups (or equivalently: $d$-parameter $C_{0}$-semigroups) over a Hilbert space for $d \in \mathbb{N}$. In the literature (\textit{cf.} [29, 26, 27, 23, 18, 25]), conditions are provided to classify the existence of unitary and regular unitary dilations. Some of these conditions require inspecting values of the semigroups, some provide only sufficient conditions, and others involve verifying sophisticated properties of the generators. By focussing on semigroups with bounded generators, we establish a simple and natural condition on the generators, \textit{viz…
All unitary perturbations of a given unitary operator $U$ by finite rank $d$ operators with fixed range can be parametrized by $(d\times d)$ unitary matrices $Γ$; this generalizes unitary rank one ($d=1$) perturbations, where the Aleksandrov--Clark family of unitary perturbations is parametrized by the scalars on the unit circle $\mathbb{T}\subset\mathbb{C}$. For a purely contractive $Γ$ the resulting perturbed operator $T_Γ$ is a contraction (a completely non-unitary contraction under the natural assumption about cyclicity of the range), so they admit the functional model. In this paper we in…
We consider a certain class of Herglotz-Nevanlinna matrix-valued functions which can be realized as the Weyl-Titchmarsh matrix-valued function of some symmetric operator and its self-adjoint extension. New properties of Weyl -Titchmarsh matrix-valued functions as well as a new version of the functional model in such realizations are presented. In the case of periodic Herglotz-Nevanlinna matrix-valued functions we provide a complete characterization of their realizations in terms of the corresponding functional model. We also obtain properties of a symmetric operator and its self-adjoint extens…
We consider a unitary cocycle or Schürmann triple on the non-commutative unitary group fixed by a complex matrix which induces an additive free white noise or an additive free Lévy process on the tensor algebra over the full Fock space. A Lévy process on a Voiculescu dual semi-group is given by a generator or Schürmann triple. We will show how a free Lévy process on the non-commutative unitary group fixed by a complex matrix can be obtained by infinitesimally convolving the additive free white noise.
We show that an operator valued $α$-completely positive map on a group G is given by a unitary representation of G on a Krein space which satisfies some condition. Moreover, two unitary equivalent such unitary representations define the same α-completely positive map. Also we introduce a pre-order relation on the collection of α-completely positive maps on a group and we characterize this relation in terms of the unitary representation associated to each map.
Phase-Retrieval in Shift-Invariant Spaces with Gaussian Generator
本文研究高斯生成元平移不变空间中函数的相位恢复问题，证明实值函数在 Beurling 密度大于 2 的分离集上可恢复至多相差一个符号，并给出复值函数的所有可能解。
Remarks on positive definite functions on a group
本文利用正分块矩阵研究群上算子值正定函数，给出Brehmer正性的新证明，分类有限群上的交换酉表示，并刻画幂保持正定或酉性的条件。
Matrix-Valued Truncated Toeplitz Operators: Properties and Applications
本文研究矩阵值截断Toeplitz算子的性质，将其应用于Nevanlinna-Pick插值问题，并刻画其酉部分、模型空间不变性及可逆性。
In this work the problem about an existence of non-measurable automorphisms of Lie groups finite and as well infinite dimensional over the field of real numbers and also over the non-archimedean local fields is investigated. Non-measurability of automorphisms is considered relative to real-valued measures and also measures with values in non-archimedean local fields. Their existence is proved and a procedure for their construction is given. Their application for a construction of non-measurable irreducible unitary representations is demonstrated.
The operator-valued multiplier theorems in weighted abstract Besov spaces are studied. These results permit us to show embedding theorems in weighted Besov-Lions type spaces. The most regular class of interpolation space is found such that the mixed differential operator is bounded from Besov-Lions space to this and Ehrling-Nirenberg-Gagliardo type sharp estimates are established. By using these results the separability properties of degenerate differential operators are studied. Especially, we prove that the associated differential operators are positive and also are generators of analytic se…
Pseudo-differential operator equations with parameter are studied. Uniform separability properties and resolvent estimates are obtained in terms of fractional derivatives. Moreover, maximal regularity properties of the pseudo-differential abstract parabolic equation are established. Particularly, it is proven that the operators generated by these pseudo-differential equations are positive and aso are generators of analytic semigroups. As an application, the anisotropic parameter dependent pseudo-differential equations and the system of pseudo-differential equations are studied
We give sufficient conditions on an operator space $E$ and on a semigroup of operators on a von Neumann algebra $M$ to obtain a bounded analytic or a $R$-analytic semigroup $(T_t \otimes Id_E)_{t \geq 0}$ on the vector valued noncommutative $L^p$-space $L^p(M,E)$. Moreover, we give applications to the $H^\infty(Σ_θ)$ functional calculus of the generators of these semigroups, generalizing some earlier work of M. Junge, C. Le Merdy and Q. Xu.
We discuss the spectral decomposition of the hypergeometric differential operators on the line $\mathrm{Re}\, z=1/2$. Such operators arise in the problem of decomposition of tensor products of unitary representations of the universal covering of the group $SL(2\,{\mathbb R}$. Our main purpose is a search of natural bases in generalized eigenspaces and variants of the inversion formula.
We study semigroups of bounded operators on a Banach space such that the members of the semigroup are continuous with respect to various weak topologies and we give sufficient conditions for the generator of the semigroup to be closed with respect to the topologies involved. The proofs of these results use the Laplace transforms of the semigroup. Thus we first give sufficient conditions for Pettis integrability of vector valued functions with respect to scalar measures.
Spline wavelet tight frames of Ron-Shen have been used widely in frame based image analysis and restorations. However, except for the tight frame property and the approximation order of the truncated series, there are few other properties of this family of spline wavelet tight frames to be known. This paper is to present a few new properties of this family that will provide further understanding of it and, hopefully, give some indications why it is efficient in image analysis and restorations. In particular, we present a recurrence formula of computing generators of higher order spline wavelet…
We present an improved version of commutator methods for unitary operators under a weak regularity condition. Once applied to a unitary operator, the method typically leads to the absence of singularly continuous spectrum and to the local finiteness of point spectrum. Large families of locally smooth operators are also exhibited. Half of the paper is dedicated to applications, and a special emphasize is put on the study of cocycles over irrational rotations. It is apparently the first time that commutator methods are applied in the context of rotation algebras, for the study of their generator…
The spectral problem for self-adjoint extensions is studied using the machinery of boundary triplets. For a class of symmetric operators having Weyl functions of a special type we calculate explicitly the spectral projections in the form of operator-valued integrals. This allows one to give a constructive proof of the fact that, in certain intervals, the resulting self-adjoint extensions are unitarily equivalent to a certain parameterizing operator acting in a smaller space, and one is able to provide an explicit form the associated unitary transform. Applications to differential operators on …
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