Multiplication operators on the Bloch space of bounded homogeneous domains

On irreducibility of a certain class of homogeneous operators obtained from quotient modules

Let $ Ω\subset \mathbb{C}^m $ be an open, connected and bounded set and $\mathcal{A}(Ω)$ be a function algebra of holomorphic functions on $Ω$. Suppose that $\mathcal{M}_q$ is the quotient Hilbert module obtained from a submodule of functions in a Hilbert module $\mathcal{M}$ vanishing to order $k$ along a smooth irreducible complex analytic set $\mathcal{Z}\subsetΩ$ of codimension at least $2$. In this article, we prove that the compression of the multiplication operators onto $\mathcal{M}_q$ is homogeneous with respect to a suitable subgroup of the automorphism group Aut$(Ω)$ of $Ω$ dependin…

酉群下齐次的乘法算子交换元组

Commuting Tuple of Multiplication Operators Homogeneous under the Unitary Group

该研究针对酉群下齐次的乘法算子交换元组，给出其与再生核希尔伯特空间上坐标乘法酉等价的条件，分类了两类拟不变核，借助SU(d)的表示性质得到算子有界性等的显式判据。

Carleson measures, Riemann-Stieltjes and multiplication operators on a general family of function spaces

Let $μ$ be a nonnegative Borel measure on the unit disk of the complex plane. We characterize those measures $μ$ such that the general family of spaces of analytic functions, $F(p,q,s)$, which contain many classical function spaces, including the Bloch space, $BMOA$ and the $Q_s$ spaces, are embedded boundedly or compactly into the tent-type spaces $T^{\infty}_{p,s}(μ)$. The results are applied to characterize boundedness and compactness of Riemann-Stieltjes operators and multiplication operators on $F(p,q,s)$.

On the isometric composition operators on the Bloch space in $\mathbb{C}^n$

Let $φ$ be a holomorphic self-map of a bounded homogeneous domain $D$ in $\mathbb{C}^n$. In this work, we show that the composition operator $C_φ: f\mapsto f\circ φ$ is bounded on the Bloch space $\mathcal{B}$ of the domain and provide estimates on its operator norm. We also give a sufficient condition for $φ$ to induce an isometry on $\cal{B}$. This condition allows us to construct non-trivial examples of isometric composition operators in the case when $D$ has the unit disk as a factor. We then obtain some necessary conditions for $C_φ$ to be an isometry on $\cal{B}$ when $D$ is a Cartan cla…

On a discrete transform of homogeneous decomposition spaces

We introduce almost diagonal matrices in the setting of (anisotropic) discrete homogeneous Triebel-Lizorkin type spaces and homogeneous modulation spaces, and it is shown that the class of almost diagonal matrices is closed under matrix multiplication. We then connect the results to the continuous setting and show that the "change of frame" matrix for a pair of time-frequency frames, with suitable decay properties, is almost diagonal. As an application of this result, we consider a construction of compactly supported frame expansions for homogeneous decomposition spaces of Triebel-Lizorkin typ…

$\mathbb K$-homogeneous tuple of operators on bounded symmetric domains

Let $Ω$ be an irreducible bounded symmetric domain of rank $r$ in $\mathbb C^d.$ Let $\mathbb K$ be the maximal compact subgroup of the identity component $G$ of the biholomorphic automorphism group of the domain $Ω$. The group $\mathbb K$ consisting of linear transformations acts naturally on any $d$-tuple $\boldsymbol T=(T_1,\ldots, T_d)$ of commuting bounded linear operators. If the orbit of this action modulo unitary equivalence is a singleton, then we say that $\boldsymbol T$ is $\mathbb{K}$-homogeneous. In this paper, we obtain a model for all $\mathbb{K}$-homogeneous $d$-tuple $\boldsym…

A Family of Homogeneous Operators In The Cowen-Douglas Class Over The Poly-disc

We construct a large family of positive-definite kernels $K: \mathbb{D}^n\times \mathbb{D}^n \to \mbox{M} (r, \mathbb C)$, holomorphic in the first variable and anti-holomorphic in the second, that are quasi-invariant with respect to the subgroup $\mbox{Möb} \times\cdots\times \mbox{Möb}$ ($n$ times) of the bi-holomorphic automorphism group of $\mathbb{D}^n$. The adjoint of the $n$ - tuples of multiplication operators by the co-ordinate functions on the Hilbert spaces $\mathcal H_K$ determined by $K$ is then homogeneous with respect to this subgroup. We show that these $n$ - tuples are irreduc…

Multiplication between elements in Martingale Hardy spaces and their duals

In this paper, we establish continuous bilinear decompositions that arise in the study of products between elements in martingale Hardy spaces $ H^p\ (0<p\leqslant 1) $ and functions in their dual spaces. Our decompositions are based on martingale paraproducts. As a consequence of our work, we also obtain analogous results for dyadic martingales on spaces of homogeneous type equipped with a doubling measure.

The orbit of a bounded operator under the Möbius group modulo similarity equivalence

Let Möb denote the group of biholomorphic automorphisms of the unit disc and $(\mbox{Möb} \cdot T)$ be the orbit of a Hilbert space operator $T$ under the action of Möb. If the quotient $(\mbox{Möb} \cdot T)/\sim$, where $\sim$ is the similarity between two operators is a singleton, then the operator $T$ is said to be weakly homogeneous. In this paper, we obtain a criterion to determine if the operator $M_z$ of multiplication by the coordinate function $z$ on a reproducing kernel Hilbert space is weakly homogeneous. We use this to show that there exists a Möbius bounded weakly homogeneous oper…

Homogeneous algebras via heat kernel estimates

We study homogeneous Besov and Triebel--Lizorkin spaces defined on doubling metric measure spaces in terms of a self-adjoint operator whose heat kernel satisfies Gaussian estimates together with its derivatives. When the measure space is a smooth manifold and such operator is a sum of squares of smooth vector fields, we prove that their intersection with $L^\infty$ is an algebra for pointwise multiplication. Our results apply to nilpotent Lie groups and Grushin settings.

Homogeneous operators on Hilbert spaces of holomorphic functions -- I

In this paper we construct a large class of multiplication operators on reproducing kernel Hilbert spaces which are {\em homogeneous} with respect to the action of the Möbius group consisting of bi-holomorphic automorphisms of the unit disc $\mathbb D$. For every $m \in \N$ we have a family of operators depending on $m+1$ positive real parameters. The kernel function is calculated explicitly. It is proved that each of these operators is bounded, lies in the Cowen-Douglas class of $\mathbb D$ and is irreducible. These operators are shown to be mutually pairwise unitarily inequivalent.

Spectral Analysis for Perturbed Operators on Carnot Groups

Let $\G$ be a Carnot group of homogeneous dimension $M$ and $Δ$ its horizontal sublaplacian. For $α\in(0,M)$ we show that operators of the form $H_α:=(-Δ)^α+V$ have no singular spectrum, under generous assumptions on the multiplication operator $V$. The proof is based on commutator methods and Hardy inequalities.