Shift invariance and reflexivity of compressions of multiplication operators

Commutants and Reflexivity of Multiplication tuples on Vector-valued Reproducing Kernel Hilbert Spaces

Motivated by the theory of weighted shifts on directed trees and its multivariable counterpart, we address the question of identifying commutant and reflexivity of the multiplication $d$-tuple $\mathscr M_z$ on a reproducing kernel Hilbert space $\mathscr H$ of $E$-valued holomorphic functions on $Ω$, where $E$ is a separable Hilbert space and $Ω$ is a bounded domain in $\mathbb C^d$ admitting bounded approximation by polynomials. In case $E$ is a finite dimensional cyclic subspace for $\mathscr M_z$, under some natural conditions on the $B(E)$-valued kernel associated with $\mathscr H$, the c…

Weighted shifts on directed trees. Their multiplier algebras, reflexivity and decompositions

We study bounded weighted shifts on directed trees. We show that the set of multiplication operators associated with an injective weighted shift on a rooted directed tree coincides with the WOT/SOT closure of the set of polynomials of the weighted shift. From this fact we deduce reflexivity of those weighted shifts on rooted directed trees whose all path-induced spectral-like radii are positive. We show that weighted shifts with positive weights on rooted directed trees admit a Wold-type decomposition. We prove that the pairwise orthogonality of the factors in the decomposition is equivalent t…

Reflexivity of the translation-dilation algebras on L^2(R)

The hyperbolic algebra A_h, studied recently by Katavolos and Power, is the weak star closed operator algebra on L^2(R) generated by H^\infty(R), as multiplication operators, and by the dilation operators V_t, t \geq 0, given by V_t f(x) = e^{t/2} f(e^t x). We show that A_h is a reflexive operator algebra and that the four dimensional manifold Lat A_h (with the natural topology) is the reflexive hull of a natural two dimensional subspace.

Cartan域上的Cartan等距与Toeplitz算子

Cartan Isometries and Toeplitz Operators on Cartan domains

本文利用Jordan三重行列式刻画经典Cartan域的Shilov边界，由此给出Cartan等距的内在特征，并研究其在双全纯自同构群下的不变性、Toeplitz算子的Brown-Halmos型条件、紧算子性质及自反性。

Near invariance and symmetric operators

Let $S$ be a subspace of $L^2 (\bm{R})$. We show that the operator $M$ of multiplication by the independent variable has a simple symmetric regular restriction to $S$ with deficiency indices $(1,1)$ if and only if $S = u h K^{2}_θ$ is a nearly invariant subspace, with $θ$ a meromorphic inner function vanishing at $i$. Here $u$ is unimodular, $h$ is an isometric multiplier of $K^{2}_θ$ into $H^2$ and $H^2$ is the Hardy space of the upper half plane. Our proof uses the dilation theory of completely positive maps.

Analyticity, rank one perturbations and the invariance of the left spectrum

We address the question of the analyticity of a rank one perturbation of an analytic operator. If $\mathscr M_z$ is the bounded operator of multiplication by $z$ on a functional Hilbert space $\mathscr H_κ$ and $f \in \mathscr H$ with $f(0)=0,$ then $\mathscr M_z + f \otimes 1$ is always analytic. If $f(0) \neq 0,$ then the analyticity of $\mathscr M_z + f \otimes 1$ is characterized in terms of the membership to $\mathscr H_κ$ of the formal power series obtained by multiplying $f(z)$ by $\frac{1}{f(0)-z}.$ As an application, we discuss the problem of the invariance of the left spectrum under …