We study the rigidity of the orthogonal projections $P_{n}:H^2(\mathbb{D}^2)\rightarrow H_{n}$, where $H^{2}(\mathbb{D}^{2})=\bigoplus_{n=0}^{\infty}H_{n}$ and each $H_{n}=H^{2}(z)w^{n}$ is a reducing subspace for $T_{z}$, and we derive properties of the minimal invariant subspaces of $T_{z}^{*}$. Our main result establishes that the nonzero projections of a minimal invariant subspace onto the one-variable Hardy space share a common $t_z^*$-invariant structure. Furthermore, we relate the results obtained to an approach to the invariant subspace problem.
Let $\mathcal{H}$ be the separable, infinite-dimensional complex Hilbert space. In the first paper of this series, we introduced, by means of the Rosenblum operators, the shift representation operators $K_x=\sum_{n=0}^{\infty}T^nx\otimes e_n$, which relate an operator $T$ with $r(T)<1$ to the unilateral shift $S$ and r...
Jun-Sheng Fang, Bing-Zhe Hou, Chun-Lan Jiang et al.· 0 citations
In this paper, we establish necessary conditions for the minimality of infinite-dimensional $T_z^*$-invariant subspaces of $H^2(\mathbb D^2)$. More precisely, we prove that if $M\subset H^2(\mathbb D^2)$ is an infinite-dimensional, minimal $T_z^*$-invariant subspace, then, for every $n\in\mathbb N_0$, either $\overline...
Let $\mathcal{H}$ be a complex Hilbert space, and let $T=H+iK\in\mathcal{B}(\mathcal{H})$ be the Cartesian decomposition of $T$, where $H=H^{*}$ and $K=K^{*}$. We introduce the $L^2$-Cartesian numerical radius and study its basic properties. We compare it with the classical numerical radius $\omega(T)$. In particular,...
W. Audeh, F. Kıttaneh, H. Moradi et al.· 0 citations
We consider $n\times n$ covariance matrices $M=\frac{1}{n}XX^*$ where $X=(x_{i,j})$ is a matrix whose entries are independent complex random variables with $\mathbb{E}(x_{i,j})=0$ and $\mathbb{E}(|x_{i,j}|^2)=1$. We derive a $\frac{1}{n}$ expansion of the mixed moments, $\frac{1}{n}\mathbb{E}(\Tr(M^{(r_1)}\cdots M^{(r_...
We study orthogonal polynomials of $d = d_1+d_2$ variables with respect to a wrapped product weight function ${\bm W}({\bm x},{\bm y}) = W_1({\bm x}/\rho({\bm y})) W_2(\bm y)$ for $(\bm x, \bm y) \in \mathbb{R}^{d_1} \times \mathbb{R}^{d_2}$, where $\rho$ is either linear or the square root of a nonnegative quadratic p...
We prove for the Euclidean space $\left(\mathbb{R}^n, \|.\|_2\right)$ the existence of subsets $A^n \subseteq \mathbb{R}^n$ with $d_H(\mathbb{R}^n, A^n)$ arbitrarily small, such that $d_{GH}(\mathbb{R}^n, A^n) = \sqrt{\frac{n+1}{2n}}d_{H}(\mathbb{R}^n, A^n)$, showing that the lower bound $d_{GH}(\mathbb{R}^n, A^n) \geq...
Paul Schott· 0 citations
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