Let $\mathscr{H}$ be a complex Hilbert space, $A$ a positive operator with closed range in $ℬ\left(\mathscr{H}\right)$ and ${ℬ}_A\left(\mathscr{H}\right)$ the sub-algebra of $ℬ\left(\mathscr{H}\right)$ of all $A$-self-adjoint operators. Assume $\phi :{ℬ}_A\left(\mathscr{H}\right)$ onto itself is a linear continuous map. This paper shows that if $\phi$ preserves $A$-unitary operators such that $\phi \left(I\right)=P$ then $\psi$ defined by $\psi \left(T\right)=P\phi \left(PT\right)$ is a homomorphism or an anti-homomorphism and $\psi \left(T^{♯}\right)=\psi {\left(T\right)}^{♯}$ for all $T\in {ℬ}_A\left(\mathscr{H}\right)$, where $P=A^{+}A$ and $A^{+}$ is the Moore-Penrose inverse of $A$. A similar result is also true if $\phi$ preserves $A$-quasi-unitary operators in both directions such that there...

Let A be a unital C*-algebra. Denote by P the space of selfadjoint projections of A. We study the relationship between P and the spaces of projections $P_a$ determined by the different involutions ${}_a$ induced by positive invertible elements a ∈ A. The maps $\phi :P\to P_a$ sending p to the unique $q\in P_a$ with the same range as p and ${\Omega }_a:P_a\to P_a$ sending q to the unitary part of the polar decomposition of the symmetry 2q-1 are shown to be diffeomorphisms. We characterize the pairs of idempotents q,r ∈ A with ||q-r|| <...

We show that the essential spectral radius ${\varrho }_e\left(T\right)$ of T ∈ B(H) can be calculated by the formula ${\varrho }_e\left(T\right)$ = inf${ℱ}_{♯·♯}\left(XT{X}^{-1}\right)$: X an invertible operator, where ${ℱ}_{♯·♯}\left(T\right)$ is a Φ₁-perturbation function introduced by Mbekhta [J. Operator Theory 51 (2004)]. Also, we show that if ${}_{♯·♯}\left(T\right)$ is a Φ₂-perturbation function [loc. cit.] and if T is a Fredholm operator, then $dist(0,{\sigma }_e\left(T\right))$ = sup${}_{♯·♯}\left(XT{X}^{-1}\right)$: X an invertible operator.

Let $V$ be a unitary space. For an arbitrary subgroup $G$ of the full symmetric group $S_m$ and an arbitrary irreducible unitary representation $\Lambda$ of $G$, we study the generalized symmetry class of tensors over $V$ associated with $G$ and $\Lambda$. Some important properties of this vector space are investigated.

Let $L\left(H\right)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$. For $A,B\in L\left(H\right)$, the generalized derivation ${\delta }_{A,B}$ and the elementary operator ${\Delta }_{A,B}$ are defined by ${\delta }_{A,B}\left(X\right)=AX-XB$ and ${\Delta }_{A,B}\left(X\right)=AXB-X$ for all $X\in L\left(H\right)$. In this paper, we exhibit pairs $(A,B)$ of operators such that the range-kernel orthogonality of ${\delta }_{A,B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of ${\Delta }_{A,B}$ with respect to the wider class of unitarily invariant...

We show that for a linear space of operators $ℳ\subseteq ℬ({\mathscr{H}}_1,{\mathscr{H}}_2)$ the following assertions are equivalent. (i) $ℳ$ is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map $\Psi =({\psi }_1,{\psi }_2)$ on a bilattice $\mathrm{Bil}\left(ℳ\right)$ of subspaces determined by $ℳ$ with $P\le {\psi }_1(P,Q)$ and $Q\le {\psi }_2(P,Q)$ for any pair $(P,Q)\in \mathrm{Bil}\left(ℳ\right)$, and such that an operator $T\in ℬ({\mathscr{H}}_1,{\mathscr{H}}_2)$ lies in $ℳ$ if and only if ${\psi }_2(P,Q)T{\psi }_1(P,Q)=0$ for all $(P,Q)\in \mathrm{Bil}\left(ℳ\right)$. This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.

Let $𝕍$ be a separable real Hilbert space, $k\in \mathbb{N}$ with $k<dim𝕍$, and let $B$ be convex and closed in $𝕍$. Let $𝒫$ be a collection of linear $k$-subspaces of $𝕍$. A point $w\in B$ is called exposed by $𝒫$ if there is a $P\in 𝒫$ so that $(w+P)\cap B=\left\{w\right\}$. We show that, under some natural conditions, $B$ can be reconstituted as the convex hull of the closure of all its exposed by $𝒫$ points whenever $𝒫$ is dense and $G_{\delta }$. In addition, we discuss the question when the set of exposed by some $𝒫$ points forms a $G_{\delta }$-set.

Let $X$ be a Banach space, $ℬ\left(X\right)$ the algebra of bounded linear operators on $X$ and $(J,\parallel ·{\parallel }_J)$ an admissible Banach ideal of $ℬ\left(X\right)$. For $T\in ℬ\left(X\right)$, let $L_{J,T}$ and $R_{J,T}\in ℬ\left(J\right)$ denote the left and right multiplication defined by $L_{J,T}\left(A\right)=TA$ and $R_{J,T}\left(A\right)=AT$, respectively. In this paper, we study the transmission of some concepts related to recurrent operators between $T\in ℬ\left(X\right)$, and their elementary operators $L_{J,T}$ and $R_{J,T}$. In particular, we give necessary and sufficient conditions for $L_{J,T}$ and $R_{J,T}$ to be sequentially recurrent. Furthermore, we prove that $L_{J,T}$ is recurrent...

We determine in ${ℝ}^n$ the form of curves $C$ corresponding to strictly monotone functions as well as the components of affine connections $\nabla$ for which any image of $C$ under a compact-free group $\Omega$ of affinities containing the translation group is a geodesic with respect to $\nabla$. Special attention is paid to the case that $\Omega$ contains many dilatations or that $C$ is a curve in ${ℝ}^3$. If $C$ is a curve in ${ℝ}^3$ and $\Omega$ is the translation group then we calculate not only the components of the curvature and the Weyl...

Let $Y$ be a hyperbolic surface and let $\phi$ be a Laplacian eigenfunction having eigenvalue $-1/4-{\tau }^2$ with $\tau >0$. Let $N\left(\phi \right)$ be the set of nodal lines of $\phi$. For a fixed analytic curve $\gamma$ of finite length, we study the number of intersections between $N\left(\phi \right)$ and $\gamma$ in terms of $\tau$. When $Y$ is compact and $\gamma$ a geodesic circle, or when $Y$ has finite volume and $\gamma$ is a closed horocycle, we prove that $\gamma$ is “good” in the sense of [TZ]. As a result, we obtain that the number of intersections between $N\left(\phi \right)$ and $\gamma$ is $O\left(\tau \right)$. This bound is...