Let $H$ be a complex Hilbert space and let ${\mathcal G}_{k}(H)$ be the Grassmannian formed by $k$-dimensional subspaces of $H$. Suppose that $\dim H>2k$ and $f$ is an orthogonality preserving injective transformation of ${\mathcal G}_{k}(H)$, i.e. for any orthogonal $X,Y\in {\mathcal G}_{k}(H)$ the images $f(X),f(Y)$ are orthogonal. If $\dim H=n$ is finite, then $n=mk+i$ for some integers $m\ge 2$ and $i\in \{0,1,\dots,k-1\}$ (for $i=0$ we have $m\ge 3$). We show that $f$ is a bijection induced by a unitary or anti-unitary operator if $i\in \{0,1,2,3\}$ or $m\ge i+1\ge 5$; in particular, the statement holds for $k\in \{1,2,3,4\}$ and, if $k\ge 5$, then there are precisely $(k-4)(k-3)/2$ values of $n$ such that the above condition is not satisfied. As an application, we obtain a result concerning the case when $H$ is infinite-dimensional.
We consider the standard quantum logic ${\mathcal L}(H)$ associated to a complex Hilbert space $H$, i.e. the lattice of closed subspaces of $H$ together with the orthogonal complementation. The orthogonality and compatibility relations are defined for any logic. In the standard quantum logic, they have a simple interpretation in terms of operator theory. For example, two closed subspaces (propositions in the logic ${\mathcal L}(H)$) are compatible if and only if the projections on these subspaces commute. We present both classical and more resent results on transformations of ${\mathcal L}(H)$…
We show that the transformations of Grassmannians (of complex Hilbert spaces) induced by linear or conjugate-linear isometries can be characterized as transformations preserving some of principal angles (corresponding to the orthogonality, adjacency and ortho-adjacency relations).
Let $H$ be a separable complex Hilbert space. Denote by ${\mathcal G}_{\infty}(H)$ the Grassmannian consisting of closed linear subspaces with infinite dimension and codimension. This Grassmannian is partially ordered by the inclusion relation. We show that every continuous order preserving bijective transformation of ${\mathcal G}_{\infty}(H)$ is induced by an invertible bounded semi-linear operator.
In this paper we present results concerning orthogonality in Hilbert $C^*$-modules. Moreover, for a $C^*$-algebra $\mathscr{A}$, we prove theorems concerning the multi-$\mathscr{A}$-linearity and its preservation by $\mathscr{A}$-linear operators. New version of solution of the orthogonality equation on Hilbert $C^*$-modules and mappings preserving orthogonality are also investigated.
Let $H$ be a separable real Hilbert space. Denote by ${\mathcal G}_{\infty}(H)$ the Grassmannian consisting of closed subspaces with infinite dimension and codimension. This Grassmannian is partially ordered by the inclusion relation. We show that every order preserving transformation of ${\mathcal G}_{\infty}(H)$ can be extended to an automorphism of the lattice of closed subspaces of $H$. It follows from Mackey's result \cite{Mackey} that automorphisms of this lattice are induced by invertible bounded linear operators.
As a partial generalisation of the Uhlhorn theorem to Hilbert $C^*$-modules, we show in this article that the module structure and the orthogonality structure of a Hilbert $C^*$-module determine its Hilbert $C^*$-module structure. In fact, we have a more general result as follows. Let $A$ be a $C^*$-algebra, $E$ and $F$ be Hilbert $A$-modules, and $I_E$ be the ideal of $A$ generated by $\{\langle x,y\rangle_A: x,y\in E\}$. If $Φ: E\to F$ is an $A$-module map, not assumed to be bounded but satisfying $$ \langle Φ(x),Φ(y)\rangle_A\ =\ 0\quad\text{whenever}\quad\langle x,y\rangle_A\ =\ 0, $$ then…
In the first part of the paper, we use states on $C^*$-algebras in order to establish some equivalent statements to equality in the triangle inequality, as well as to the parallelogram identity for elements of a pre-Hilbert $C^*$-module. We also characterize the equality case in the triangle inequality for adjointable operators on a Hilbert $C^*$-module. Then we give certain necessary and sufficient conditions to the Pythagoras identity for two vectors in a pre-Hilbert $C^*$-module under the assumption that their inner product has negative real part. We introduce the concept of Pythagoras orth…
We introduce a notion of approximate orthogonality preserving mappings between Hilbert $C^*$-modules. We define the concept of $(δ, \varepsilon)$-orthogonality preserving mapping and give some sufficient conditions for a linear mapping to be $(δ, \varepsilon)$-orthogonality preserving. In particular, if $\mathscr{E}$ is a full Hilbert $\mathscr{A}$-module with $\mathbb{K}(\mathscr{H})\subseteq \mathscr{A} \subseteq \mathbb{B}(\mathscr{H})$ and $T, S:\mathscr{E}\longrightarrow \mathscr{E}$ are two linear mappings satisfying $\big|\langle Sx, Sy\rangle\big| = \|S\|^2\,|\langle x, y\rangle|$ for …
Due to the corresponding fact concerning Hilbert spaces, it is natural to ask if the linearity and the orthogonality structure of a Hilbert $C^*$-module determine its $C^*$-algebra-valued inner product. We verify this in the case when the $C^*$-algebra is commutative (or equivalently, we consider a Hilbert bundle over a locally compact Hausdorff space). More precisely, a $\mathbb{C}$-linear map $θ$ (not assumed to be bounded) between two Hilbert $C^*$-modules is said to be "orthogonality preserving" if $\left<θ(x),θ(y)\right> =0$ whenever $\left<x,y\right> =0$. We prove that if $θ$ is an ortho…
Let $A$ be a $C^*$-algebra. Let $E$ and $F$ be Hilbert $A$-modules with $E$ being full. Suppose that $θ: E\to F$ is a linear map preserving orthogonality, i.e., $<θ(x), θ(y) > = 0$ whenever $<x, y > = 0$. We show in this article that if, in addition, $A$ has real rank zero, and $θ$ is an $A$-module map (not assumed to be bounded), then there exists a central positive multiplier $u\in M(A)$ such that $<θ(x), θ(y) > = u < x, y>$ ($x,y\in E$). In the case when $A$ is a standard $C^*$-algebra, or when $A$ is a $W^*$-algebra containing no finite type II direct summand, we also obtain the same concl…
In this paper, the concept of Birkhoff--James orthogonality of operators on a Hilbert space is generalized when a semi-inner product is considered. More precisely, for linear operators $T$ and $S$ on a complex Hilbert space $\mathcal{H}$, a new relation $T\perp^B_A S$ is defined if $T$ and $S$ are bounded with respect to the seminorm induced by a positive operator $A$ satisfying ${\|T + γS\|}_A\geq {\|T\|}_A$ for all $γ\in \mathbb{C}$. We extend a theorem due to R. Bhatia and P. Šemrl, by proving that $T\perp^B_A S$ if and only if there exists a sequence of $A$-unit vectors $\{x_n\}$ in $\math…
We show that if the conclusion of the well known Stampacchia Theorem, on variational inequalities, holds on a Banach space X, then X is isomorphic to a Hilbert space. Motivated by this we obtain a relevant result concerning self-dual Banach spaces and investigate some connections between existing notions of orthogonality and self-duality. Moreover, we revisit the notion of the cosine of a linear operator and show that it can be used to characterize Hilbert space structure. Finally, we present some consequences of our results to quadratic forms and to evolution triples.
Let $\mathbb{B}(\mathcal{H})$ denote the $C^{\ast}$-algebra of all bounded linear operators on a Hilbert space $\big(\mathcal{H}, \langle\cdot, \cdot\rangle\big)$. Given a positive operator $A\in\B(\h)$, and a number $λ\in [0,1]$, a seminorm ${\|\cdot\|}_{(A,λ)}$ is defined on the set $\B_{A^{1/2}}(\h)$ of all operators in $\B(\h)$ having an $A^{1/2}$-adjoint. The seminorm ${\|\cdot\|}_{(A,λ)}$ is a combination of the sesquilinear form ${\langle \cdot, \cdot\rangle}_A$ and its induced seminorm ${\|\cdot\|}_A$. A characterization of Birkhoff--James orthogonality for operators with respect to th…
We consider homoclinic solutions for Hamiltonian systems in symplectic Hilbert spaces and generalise spectral flow formulas that were proved by Pejsachowicz and the author in finite dimensions some years ago. Roughly speaking, our main theorem relates the spectra of infinite dimensional Hamiltonian systems under homoclinic boundary conditions to intersections of their stable and unstable spaces. Our proof has some interest in its own. Firstly, we extend a celebrated theorem by Cappell, Lee and Miller about the classical Maslov index in $\mathbb{R}^{2n}$ to symplectic Hilbert spaces. Secondly, …
An equiangular tight frame (ETF) is a sequence of vectors in a Hilbert space that achieves equality in the Welch bound and so has minimal coherence. More generally, an equichordal tight fusion frame (ECTFF) is a sequence of equi-dimensional subspaces of a Hilbert space that achieves equality in Conway, Hardin and Sloane's simplex bound. Every ECTFF is a type of optimal Grassmannian code, that is, an optimal packing of equi-dimensional subspaces of a Hilbert space. We construct ECTFFs by exploiting new relationships between known ETFs. Harmonic ETFs equate to difference sets for finite abelian …
Let $\big(\mathscr{X}, \langle\cdot, \cdot\rangle\big)$ be a Hilbert $C^*$-module over a $C^*$-algebra $\mathscr{A}$ and let $\mathcal{S}(\mathscr{A})$ be the set of states on $\mathscr{A}$. In this paper, we first compute the norm derivative for elements $x$ and $y$ of $\mathscr{X}$ as follows \begin{align*} ρ_{_{+}}(x, y) = \max\Big\{\mbox{Re}\,φ(\langle x, y\rangle): \, φ\in \mathcal{S}(\mathscr{A}), φ(\langle x, x\rangle) = \|x\|^2\Big\}. \end{align*} We then apply it to characterize different concepts of orthogonality in $\mathscr{X}$. In particular, we present a simpler proof of the clas…
Let $E,F$ be Banach spaces. In the case that $F$ is reflexive we give a description for the solutions $(f,g)$ of the Banach-orthogonality equation $$\langle f(x),g(α)\rangle=\langle x,α\rangle\hspace{10mm}\forall x\in E,\forall α\in E^*,$$ where $f:E\rightarrow F,g:E^*\rightarrow F^*$ are two maps. Our result generalizes the recent result of Łukasik and Wójcik in the case that $E$ and $F$ are Hilbert spaces.
For a unital $C^*$-algebra $\mathcal A$ and a subspace $\mathcal B$ of $\mathcal A$, a characterization for a best approximation to an element of $\mathcal A$ in $\mathcal B$ is obtained. As an application, a formula for the distance of an element of $\mathcal A$ from $\mathcal B$ has been obtained, when a best approximation of that element to $\mathcal B$ exists. Further, a characterization for Birkhoff-James orthogonality of an element of a Hilbert $C^*$-module to a subspace is obtained.
A mapping $f: {\mathcal M} \to {\mathcal N}$ between Hilbert $C^*$-modules approximately preserves the inner product if \[\|< f(x), f(y)> - < x, y> \| \leq ϕ(x, y),\] for an appropriate control function $ϕ(x,y)$ and all $x, y \in {\mathcal M}$. In this paper, we extend some results concerning the stability of the orthogonality equation to the framework of Hilbert $C^*$-modules on more general restricted domains. In particular, we investigate some asymptotic behavior and the Hyers--Ulam--Rassias stability of the orthogonality equation.
Utilizing the Birkhoff--James orthogonality, we present some characterizations of the norm-parallelism for elements of $\mathbb{B}(\mathscr{H})$ defined on a finite dimensional Hilbert space, elements of a Hilbert $C^*$-module over the $C^*$-algebra of compact operators and elements of an arbitrary $C^*$-algebra. We also consider the characterization of norm parallelism problem for operators on a finite dimensional Hilbert space when the operator norm is replaced by the Schatten $p$-norm. Some applications and generalizations are discussed for certain elements of a Hilbert $C^*$-module.
The Restricted Schatten-class Grassmannian $\mathrm{Gr}_{\mathrm{res}, p}(\mathcal{H})$ as affine coadjoint orbit
University of Utrecht（乌得勒支大学）；UMR CNRS 8524 UFR de Mathématiques Laboratoire Paul Painlevé（CNRS 8524 联合研究实验室 数学学院 保罗·潘勒瓦实验室）；Institut CNRS Pauli UMI CNRS 2842（保罗·庞加莱研究所 CNRS 2842 联合国际单位）
Let $\mathcal{H}$ be a Hilbert space, and let $K(\mathcal{H})$ be the $C^*$-algebra of compact operators on $\mathcal{H}$. In this paper, we present some characterizations of the norm-parallelism for elements of a Hilbert $K(\mathcal{H})$-module by employing the Birkhoff--James orthogonality. Among other things, we present a characterization of transitive relation of the norm-parallelism for elements in a certain Hilbert $K(\mathcal{H})$-module. We also give some characterizations of the Schatten $p$-norms and the operator norm-parallelism for matrices.
Let $H$ be a complex separable Hilbert space and $B(H)$ the algebra of all bounded linear operators on $H$. In this paper, we give considerable generalizations of the inequalities for norms of commutators of normal operators. Let $S, T \in B(H)$ be positive normal operators with the cartesian decomposition $S=A+iC$ and $T=B+iD$ such that $a_{1}\leq A \leq a_{2},\, b_{1}\leq B \leq b_{2},\,c_{1}\leq C \leq c_{2}$ and $ d_{1}\leq D \leq d_{2}$ for some real numbers $a_{1},\,a_{2},\,b_{1},\,b_{2},\,c_{1},\,c_{2},\,d_{1}$ and $d_{2}$ we have shown that $\|ST-TS\|\leq \frac{1}{2}\sqrt{(a_{2}-a_{1})…
