Stephan Ramon Garcia, Warren R. Wogen
An operator $T \in B(\h)$ is complex symmetric if there exists a conjugate-linear, isometric involution $C:\h\to\h$ so that $T = CT^*C$. We provide a concrete description of all complex symmetric partial isometries. In particular, we prove that any partial isometry on a Hilbert space of dimension $\leq 4$ is complex symmetric.
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Under the mild condition of continuity at a single point we describe all the bijections of the set of all partial isometries on a Hilbert space which preserve the order and the orthogonality in both directions. Moreover, we present a natural analogue of Wigner's theorem on quantum mechanical symmetries for the set of all rank-1 partial isometries.
We give a new formulation and proof of a theorem of Halmos and Wallen on the structure of power partial isometries on Hilbert space. We then use this theorem to give a structure theorem for a finite set of partial isometries which star-commute: each operator commutes with the others and with their adjoints.
The aim of the present paper is, firstly we study the concepts of (m, (q_1, ..., q_d))- partial isometries on a Hilbert space, secondly, we introduce the notion of m- invertibility of tuples of operators as a natural generalization of the m-invertibility in single variable operators.
We study positivity in C*-modules and operator spaces using open tripotents, and an ordered version of the `noncommutative Shilov boundary'. Because of their independent interest, we also systematically study open tripotents and their properties.
Partial Isometries Between Hilbert Modules and Their Compositions
受Accardi和Lu的问题启发，研究希尔伯特空间中两部分等距乘积仍为部分等距的准则，定义新复合运算，并证明该运算无法直接推广到希尔伯特模，需转向部分定义等距类。
We prove, in some cases in term of kippenhahn curve, that if 5-by-5 partial isometry whose numerical range is a circular disc then its center is must be the origin. This gives a partial affirmative answer of the Conjecture 5.1. of [H. l. Gau et al., Linear and Multilinear Algebra, 64 (1) 2016, 14--35.], for the five dimensional case.
On $m$-partial isometries: spectra, weighted shifts, and similarity
本文研究Hilbert空间上的$m$-部分等距算子，建立结构和谱结果，刻画$m$-部分等距加权移位，并探讨与$m$-等距和$m$-部分等距的相似性。
We study self-adjoint semigroups of partial isometries on a Hilbert space. These semigroups coincide precisely with faithful representations of abstract inverse semigroups. Groups of unitary operators are specialized examples of self-adjoint semigroups of partial isometries. We obtain a general structure result showing that every self-adjoint semigroup of partial isometries consists of "generalized weighted composition" operators on a space of square-integrable Hilbert-space valued functions. If the semigroup is irreducible and contains a compact operator then the underlying measure space is p…
On circular Kippenhahn curves and the Gau-Wang-Wu conjecture about nilpotent partial isometries
本文研究具有圆状Kippenhahn曲线的线性算子，探讨其圆性性质，并通过构造更广的算子类推翻Gau-Wang-Wu猜想，提出数值验证旋转不变性的方法。
Let $\mathcal S$ be a semigroup of partial isometries acting on a complex, infinite-dimensional, separable Hilbert space. In this paper we seek criteria which will guarantee that the selfadjoint semigroup $\mathcal T$ generated by $\mathcal S$ consists of partial isometries as well. Amongst other things, we show that this is the case when the set of final projections of elements of $\mathcal S$ generates an abelian von Neumann algebra of uniform finite multiplicity.
Let $\mathcal{M}$ be an atomless semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbert space $H$ equipped with a semifinite faithful normal trace $τ$. Let $E(\mathcal{M},τ) $ be a symmetric operator space affiliated with $ \mathcal{M} $, whose norm is order continuous and is not proportional to the Hilbertian norm $\left\|\cdot\right\|_2$ on $L_2(\mathcal{M},τ)$. We obtain general description of all bounded hermitian operators on $E(\mathcal{M},τ)$. This is the first time that the description of he…
We investigate some bounded linear operators T on a Hilbert space which satisfy the condition |T | less or equal to |ReT |. We describe the maximum invariant subspace for a contraction T on which T is a partial isometry to obtain that, in certain cases, the above condition ensures that T is self-adjoint. In other words we show that the Fong-Tsui conjecture holds for partial isometries, contractive quasi-isometries, or 2-quasi-isometries, and Brownian isometries of positive covariance, or even for a more general class of operators.
Power partial isometries
本文完整刻画了幂部分等距算子完全非酉部分的约化、不变与超不变子空间，并特别给出了截断移位算子三类子空间的精确刻画。
Two statements concerning $n$-by-$n$ partial isometries are being considered: (i) these matrices are generic, if unitarily irreducible, and (ii) if nilpotent, their numerical ranges are circular disks. Both statements hold for $n\leq 4$ but fail starting with $n=5$.
We solve the problem of best approximation by partial isometries of given rank to an arbitrary rectangular matrix, when the distance is measured in any unitarily invariant norm. In the case where the norm is strictly convex, we parametrize all the solutions. In particular, this allow us to give a simple necessary and sufficient condition for uniqueness. We then apply these results to solve the global problem of approximation by partial isometries, and to extend the notion of symmetric approximation of frames introduced in M. Frank, V. Paulsen, T. Tiballi, Symmetric Approximation of frames and …
This paper studies three natural pre-orders of increasing generality on the set of all completely non-unitary partial isometries with equal defect indices. We show that the problem of determining when one partial isometry is less than another with respect to these pre-orders is equivalent to the existence of a bounded (or isometric) multiplier between two natural reproducing kernel Hilbert spaces of analytic functions. For large classes of partial isometries these spaces can be realized as the well-known model subspaces and deBranges-Rovnyak spaces. This characterization is applied to investig…
This note is concerned with isometries on the spaces of self-adjoint traceless matrices. We compute the group of isometries with respect to any unitary similarity invariant norm. This completes and extends the result of Nagy on Schatten $p$-norm isometries. Furthermore, we point out that our proof techniques could be applied to obtain an old result concerning isometries on skew-symmetric matrices.
On Kippenhahn curves of low rank partial isometries
本文研究了低秩部分等距的基尔彭哈恩曲线，探讨了其圆状成分条件及数值范围特性，并验证了相关数学猜想。
We show that ||u*u - v*v|| \leq ||u - v|| for partial isometries u and v. There is a stronger inequality if both u and v are extreme points of the unit ball of a C*-algebra, and both inequalities are sharp. If u and v are partial isometries in a C*-algebra A such that ||u - v|| < 1, then u and v are homotopic through partial isometries in A. If both u and v are extremal, then it is sufficient that ||u - v|| < 2. The constants 1 and 2 are both sharp. We also discuss the continuity points of the map which assigns to each closed range element of A the partial isometry in its canonical polar decom…
Banach Lie groupoid of partial isometries over restricted Grassmannian
本文构造了限制Grassmannian上部分等距集的微分结构，使其成为Banach Lie groupoid。
We determine when a matrix is similar to a partial isometry, refining a result of Halmos--McLaughlin.
The aim of this paper is to show that two accessible subspaces in the Loynes $\mathcal{Z}$ - space $\Hc$ are the initial and final space of a partial gramian isometry, respectively if the norm of the difference of the associated gramian selfadjoint projections is strictly less than 1.
Gau, Wang and Wu in their LAMA'2016 paper conjectured (and proved for $n\leq 4$) that an $n$-by-$n$ partial isometry cannot have a circular numerical range with a non-zero center. We prove that this statement holds also for $n=5$.
Let $n>1$ and let $\{U_{ij}\}_{1\leq i<j\leq n}$ be $n\choose 2$ commuting unitaries on a Hilbert space $\mathcal{H}$. Suppose $U_{ji}:=U^*_{ij}$, $1\leq i<j\leq n$. An n-tuple of power partial isometries $(V_1,...,V_n)$ on Hilbert space $\mathcal{H}$ is called $\mathcal{U}_n$-twisted power partial isometry with respect to $\{U_{ij}\}_{i<j}$ (or simply $\mathcal{U}_n$-twisted power partial isometry if $\{U_{ij}\}_{i<j}$ is clear from the context) if $V_i^*V_j=U_{ij}V_jV^*_i, ~~ V_iV_j=U_{ji}V_jV_i ~~\text{and}~~ V_kU_{ij}=U_{ij}V_k~~(i,j,k=1,2,...,n,~\text{and}~i\neq j).$ We prove that each $\…
In this paper, we extend the notion of orthogonality to the general elements of an absolute matrix order unit space and relate it to the orthogonality among positive elements. We introduce the notion of a partial isometry in an absolute matrix order unit space. As an application, we describe the comparison of order projections. We also discuss finiteness of order projections.
Order automorphisms of partial isometries in $M_n(\mathbb C)$
本文刻画了矩阵代数中部分等距集合上的序自同构，发现其无法由标准可逆或酉变换实现，而由 $I-(A+A^*)$ 正或负可逆的矩阵支配，结构比经典子空间自同构更复杂。
A known general program, designed to endow the quotient space ${\cal U}_{\cal A} / {\cal U}_{\cal B}$ of the unitary groups ${\cal U}_{\cal A}$, ${\cal U}_{\cal B}$ of the C$^*$ algebras ${\cal B}\subset{\cal A}$ with an invariant Finsler metric, is applied to obtain a metric for the space ${\cal I}({\cal H})$ of partial isometries of a Hilbert space ${\cal H}$. ${\cal I}({\cal H})$ is a quotient of the unitary group of ${\cal B}({\cal H})\times{\cal B}({\cal H})$, where ${\cal B}({\cal H})$ is the algebra of bounded linear operators in ${\cal H}$. Under this program, the solution of a linear …
We derive a matrix model, under unitary similarity, of an $n$-by-$n$ matrix $A$ such that $A, A^2, \ldots, A^k$ ($k\ge 1$) are all partial isometries, which generalizes the known fact that if $A$ is a partial isometry, then it is unitarily similar to a matrix of the form ${\scriptsize\left[\begin{array}{cc} 0 & B 0 & C\end{array}\right]}$ with $B^*B+C^*C=I$. Using this model, we show that if $A$ has ascent $k$ and $A, A^2, \ldots, A^{k-1}$ are partial isometries, then the numerical range $W(A)$ of $A$ is a circular disc centered at the origin if and only if $A$ is unitarily similar to a direct…
In their LAMA'2016 paper Gau, Wang and Wu conjectured that a partial isometry $A$ acting on $\mathbb{C}^n$ cannot have a circular numerical range with a non-zero center, and proved this conjecture for $n\leq 4$. We prove it for operators with $\mathrm{rank}\,A=n-1$ and any $n$. The proof is based on the unitary similarity of $A$ to a compressed shift operator $S_B$ generated by a finite Blaschke product $B$. We then use the description of the numerical range of $S_B$ as intersection of Poncelet polygons, a special representation of Blaschke products related to boundary interpolation, and an ex…
Given a row contraction of operators on Hilbert space and a family of projections on the space which stabilize the operators, we show there is a unique minimal joint dilation to a row contraction of partial isometries which satisfy natural relations. For a fixed row contraction the set of all dilations forms a partially ordered set with a largest and smallest element. A key technical device in our analysis is a connection with directed graphs. We use a Wold Decomposition for partial isometries to describe the models for these dilations, and discuss how the basic properties of a dilation depend…
Isometries of Unit Spheres of SFS-Spaces and a Geometric Approach to Tingley's Problem
本文研究面向对称空间单位球的满射等距映射，利用其几何性质为球等距映射的线性延拓提供框架，解决了满足联合Peirce性质的有限维实中性强面向对称空间的Tingley问题。
This article gives a procedure to convert a frame which is not a tight frame into a Parseval frame for the same space, with the requirement that each element in the resulting Parseval frame can be explicitly written as a linear combination of the elements in the original frame. Several examples are considered, such as a Fourier frame on a spiral. The procedure can be applied to the construction of Parseval frames for L^2(B(0,R)), the space of square integrable functions whose domain is the ball of radius R. When a finite number of measurements are used to reconstruct a signal in L^2(B(0,R)), e…
A Counterexample to the Gau--Wang--Wu Conjecture on Partial Isometries
该研究构造反例否定了Gau等提出的部分等距数值域为圆形时圆心必在原点的猜想，证明存在特定4秩部分等距，其数值域为圆心偏离原点的圆。
Let $ n\geq 2, A=(a_{ij})_{i,j=1}^{n}$ be a real symmetric matrix, $a=(a_i)_{i=1}^{n}\in \Bbb R^n.$ Consider the differential operator $D_A = \sum_{i,j=1}^n a_{ij}{\partial^2 \over \partial x_i \partial x_j}+ \sum_{i=1}^n a_i{\partial \over \partial x_i}.$ Let $E$ be a bounded domain in $\Bbb R^n,$ $p>0.$ Denote by $L_{D_A}^p(E)$ the space of solutions of the equation $D_A f=0$ in the domain $E$ provided with the $L_p$-norm. We prove that, for matrices $A,B,$ vectors $a,b,$ bounded domains $E,F,$ and every $p>0$ which is not an even integer, the space $L_{D_A}^p(E)$ is isometric to a subspace …
A family $\{T_j\}_{j\in J}$ of commuting Hilbert space operators is said to be a spherical isometry if $\sum_{j\in J}T^*_jT_j=1$ in the weak operator topology. We show that every commuting family $\Cal F$ of spherical isometries has a commuting normal extension $\hat{\Cal F}$. Moreover, if $\hat{\Cal F}$ is minimal, then there exists a natural short exact sequence $0\to\Cal C\to C^*(\Cal F)\to C^*(\hat{\Cal F})\to 0$ with a completely isometric cross-section, where $\Cal C$ is the commutator ideal in $C^*(\Cal F)$. We also show that the space of Toeplitz operators associated to $\Cal F$ is com…
Spectral Characterizations of Schatten-class perturbations of Partial isometries
研究刻画部分等距算子标量倍数的紧或Schatten类扰动的有界算子，通过\(T^*T\)本质谱等进行，给出相关充要条件及等价准则，还应用于等距算子扰动刻画、Moore - Penrose逆行为描述及闭值域算子分解。
There exist several interesting results in the literature on subnormal operator tuples having their spectral properties tied to the geometry of strictly pseudoconvex domains or to that of bounded symmetric domains in $\C^n$. We introduce a class $Ω^{(n)}$ of convex domains in $\C^n$ which, for $n \geq 2$, is distinct from the class of strictly pseudoconvex domains and the class of bounded symmetric domains and which lends itself for the application of the theories related to the abstract inner function problem and the $\bar \partial$-Neumann problem, allowing us to make a number of interesting…
Maximally monotone operators are fundamental objects in modern optimization. The main classes of monotone operators are subdifferential operators and matrices with a positive semidefinite symmetric part. In this paper, we study a nice class of monotone operators: displacement mappings of isometries of finite order. We derive explicit formulas for resolvents, Yosida approximations, and (set-valued and MoorePenrose) inverses. We illustrate our results by considering certain rational rotators and circular shift operators.
We prove the existence of the invariant subspaces of some operators in a real Banach space. For example, linear isometries have invariant subspaces
Linear isometries on the annulus: description and spectral properties
该研究完整刻画了半平面、复平面及对称环带上全纯函数空间的线性等距，提出描述环带上保单位圆全纯映射的新技术，并证明了环带线性等距的谱性质结果。
In this paper we define a notion of S-extension for a metric space and study minimality and coherence of S-extensions. We show that every S-extension can be identified with an algebraic object. We use this algebraic representation to give a complete characterization of all finite minimal S-extensions of a given finite metric space and a complete characterization of all minimal coherent S-extensions. We also define a notion of ultraextensive metric spaces and show that every countable metric space can be extended to a countable ultraextensive metric space. %As an application, we show that every…
We present geometric characterizations of the partial isometries, unitaries, and invertible operators in C*-algebras and von Neumann algebras.
We say that an operator $T \in B(H)$ is complex symmetric if there exists a conjugate-linear, isometric involution $C:H\to H$ so that $T = CT^*C$. We prove that binormal operators, operators that are algebraic of degree two (including all idempotents), and large classes of rank-one perturbations of normal operators are complex symmetric. From an abstract viewpoint, these results explain why the compressed shift and Volterra integration operator are complex symmetric. Finally, we attempt to describe all complex symmetric partial isometries, obtaining the sharpest possible statement given only t…
An operator T on Hilbert space is a 3-isometry if there exists operators B and D such that (T*)^n T^n = I+nB +n^2 D. An operator J is a Jordan operator if it the sum of a unitary U and nilpotent N of order two which commute. If T is a 3-isometry and c>0, then I-c^{-2} D + sB + s^2D is positive semidefinite for all real s if and only if T is the restriction to an invariant subspace of a Jordan operator J=U+N with the norm of N at most c. As a corollary, an analogous result for 3-symmetric operators, due to Helton and Agler, is recovered.
Motivated by importance of operator spaces contained in the set of all scalar multiples of isometries ($MI$-spaces) in a separable Hilbert space for $C^*$-algebras and E-semigroups we exhibit more properties of such spaces. For example, if an $MI$-space contains an isometry with shift part of finite multiplicity, then it is one-dimensional. We propose a simple model of a unilateral shift of arbitrary multiplicity and show that each separable subspace of a Hilbert space is the range of a shift. Also, we show that $MI$-spaces are non-symmetric, very unfriendly to multiplication, and prove a Comm…
On a class of left ideals of nest algebras
本文研究由全序偏等距族确定的巢代数左理想，刻画了其为子代数的充要条件，并证明其可分解性、有限秩算子强稠密性及方程可解条件。
The polar factor of a bounded operator acting on a Hilbert space is the unique partial isometry arising in the polar decomposition. It is well known that the polar factor might not be a best approximant to its associated operator in the set of all partial isometries, when the distance is measured in the operator norm. We show that the polar factor of an arbitrary operator $T$ is a best approximant to $T$ in the set of all partial isometries $X$ such that $\dim (\ker(X)\cap \ker(T)^\perp)\leq \dim (\ker(X)^\perp\cap \ker(T))$. We also provide a characterization of best approximations. This work…
We begin a program of generalizing basic elements of the theory of comparison, equivalence, and subequivalence, of elements in C*-algebras, to the setting of more general algebras. In particular, we follow the recent lead of Lin, Ortega, Rordam, and Thiel of studying these equivalences, etc., in terms of open projections or module isomorphisms. We also define and characterize a new class of inner ideals in operator algebras, and develop a matching theory of open partial isometries in operator ideals which simultaneously generalize the open projections in operator algebras (in the sense of the …
Let $\mathcal{E}$ and $\mathcal{F}$ be symmetrically $Δ$-normed (in particular, quasi-normed) operator spaces affiliated with semifinite von Neumann algebras $\mathcal{M}_1$ and $\mathcal{M}_2$, respectively. We establish a noncommutative version of Abramovich's theorem \cite{A1983}, which provides the general form of normal order-preserving linear operators $T:\mathcal{E} \stackrel{into}{\longrightarrow} \mathcal{F}$ having the disjointness preserving property. As an application, we obtain a noncommutative Huijsmans-Wickstead theorem \cite{Huijsmans_W}. By establishing the disjointness preser…
Let U be an open subset of R^n. Let L^2=L^2(U,dx) and H^1_0=H^1_0(U) be the standard Lebesgue and Sobolev spaces of complex-valued functions. The aim of this paper is to study the group G of invertible operators on H^1_0 which preserve the L^2-inner product. When U is bounded and the border $\partial U$ is smooth, this group acts as the intertwiner of the H^1_0 solutions of the non-homogeneous Helmholtz equation $u-Δu=f$, $u|_{\partial U}=0$. We show that G is a real Banach-Lie group, whose Lie algebra is (i times) the space of symmetrizable operators. We discuss the spectrum of operators belo…
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